:: CFUNCT_1 semantic presentation

definition
mode Complex is Element of COMPLEX ;
end;

definition
let c1 be non empty set ;
let c2, c3 be PartFunc of c1, COMPLEX ;
deffunc H1( set ) -> Element of COMPLEX = (c2 /. a1) * ((c3 /. a1) " );
func c2 / c3 -> PartFunc of a1, COMPLEX means :Def1: :: CFUNCT_1:def 1
( dom a4 = (dom a2) /\ ((dom a3) \ (a3 " {0})) & ( for b1 being Element of a1 holds
( b1 in dom a4 implies a4 /. b1 = (a2 /. b1) * ((a3 /. b1) " ) ) ) );
existence
ex b1 being PartFunc of c1, COMPLEX st
( dom b1 = (dom c2) /\ ((dom c3) \ (c3 " {0})) & ( for b2 being Element of c1 holds
( b2 in dom b1 implies b1 /. b2 = (c2 /. b2) * ((c3 /. b2) " ) ) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, COMPLEX holds
( dom b1 = (dom c2) /\ ((dom c3) \ (c3 " {0})) & ( for b3 being Element of c1 holds
( b3 in dom b1 implies b1 /. b3 = (c2 /. b3) * ((c3 /. b3) " ) ) ) & dom b2 = (dom c2) /\ ((dom c3) \ (c3 " {0})) & ( for b3 being Element of c1 holds
( b3 in dom b2 implies b2 /. b3 = (c2 /. b3) * ((c3 /. b3) " ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def1 defines / CFUNCT_1:def 1 :
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds
( b4 = b2 / b3 iff ( dom b4 = (dom b2) /\ ((dom b3) \ (b3 " {0})) & ( for b5 being Element of b1 holds
( b5 in dom b4 implies b4 /. b5 = (b2 /. b5) * ((b3 /. b5) " ) ) ) ) );

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, COMPLEX ;
deffunc H1( set ) -> Element of COMPLEX = (c2 /. a1) " ;
func c2 ^ -> PartFunc of a1, COMPLEX means :Def2: :: CFUNCT_1:def 2
( dom a3 = (dom a2) \ (a2 " {0}) & ( for b1 being Element of a1 holds
( b1 in dom a3 implies a3 /. b1 = (a2 /. b1) " ) ) );
existence
ex b1 being PartFunc of c1, COMPLEX st
( dom b1 = (dom c2) \ (c2 " {0}) & ( for b2 being Element of c1 holds
( b2 in dom b1 implies b1 /. b2 = (c2 /. b2) " ) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, COMPLEX holds
( dom b1 = (dom c2) \ (c2 " {0}) & ( for b3 being Element of c1 holds
( b3 in dom b1 implies b1 /. b3 = (c2 /. b3) " ) ) & dom b2 = (dom c2) \ (c2 " {0}) & ( for b3 being Element of c1 holds
( b3 in dom b2 implies b2 /. b3 = (c2 /. b3) " ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines ^ CFUNCT_1:def 2 :
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds
( b3 = b2 ^ iff ( dom b3 = (dom b2) \ (b2 " {0}) & ( for b4 being Element of b1 holds
( b4 in dom b3 implies b3 /. b4 = (b2 /. b4) " ) ) ) );

theorem Th1: :: CFUNCT_1:1
canceled;

theorem Th2: :: CFUNCT_1:2
canceled;

theorem Th3: :: CFUNCT_1:3
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds
( dom (b2 + b3) = (dom b2) /\ (dom b3) & ( for b4 being Element of b1 holds
( b4 in dom (b2 + b3) implies (b2 + b3) /. b4 = (b2 /. b4) + (b3 /. b4) ) ) )
proof end;

theorem Th4: :: CFUNCT_1:4
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds
( dom (b2 - b3) = (dom b2) /\ (dom b3) & ( for b4 being Element of b1 holds
( b4 in dom (b2 - b3) implies (b2 - b3) /. b4 = (b2 /. b4) - (b3 /. b4) ) ) )
proof end;

theorem Th5: :: CFUNCT_1:5
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds
( dom (b2 (#) b3) = (dom b2) /\ (dom b3) & ( for b4 being Element of b1 holds
( b4 in dom (b2 (#) b3) implies (b2 (#) b3) /. b4 = (b2 /. b4) * (b3 /. b4) ) ) )
proof end;

theorem Th6: :: CFUNCT_1:6
canceled;

theorem Th7: :: CFUNCT_1:7
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX
for b3 being Element of COMPLEX holds
( dom (b3 (#) b2) = dom b2 & ( for b4 being Element of b1 holds
( b4 in dom (b3 (#) b2) implies (b3 (#) b2) /. b4 = b3 * (b2 /. b4) ) ) )
proof end;

theorem Th8: :: CFUNCT_1:8
canceled;

theorem Th9: :: CFUNCT_1:9
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds
( dom (- b2) = dom b2 & ( for b3 being Element of b1 holds
( b3 in dom (- b2) implies (- b2) /. b3 = - (b2 /. b3) ) ) )
proof end;

Lemma8: for b1, b2 being set
for b3 being non empty set
for b4 being PartFunc of b3, COMPLEX holds
( b1 in b4 " b2 iff ( b1 in dom b4 & b4 /. b1 in b2 ) )
by PARTFUN2:44;

theorem Th10: :: CFUNCT_1:10
canceled;

theorem Th11: :: CFUNCT_1:11
canceled;

theorem Th12: :: CFUNCT_1:12
canceled;

theorem Th13: :: CFUNCT_1:13
canceled;

theorem Th14: :: CFUNCT_1:14
canceled;

theorem Th15: :: CFUNCT_1:15
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds
( dom (b2 ^ ) c= dom b2 & (dom b2) /\ ((dom b2) \ (b2 " {0})) = (dom b2) \ (b2 " {0}) )
proof end;

theorem Th16: :: CFUNCT_1:16
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds (dom (b2 (#) b3)) \ ((b2 (#) b3) " {0}) = ((dom b2) \ (b2 " {0})) /\ ((dom b3) \ (b3 " {0}))
proof end;

theorem Th17: :: CFUNCT_1:17
for b1 being non empty set
for b2 being Element of b1
for b3 being PartFunc of b1, COMPLEX holds
not ( b2 in dom (b3 ^ ) & not b3 /. b2 <> 0 )
proof end;

theorem Th18: :: CFUNCT_1:18
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds (b2 ^ ) " {0} = {}
proof end;

theorem Th19: :: CFUNCT_1:19
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds
( |.b2.| " {0} = b2 " {0} & (- b2) " {0} = b2 " {0} )
proof end;

theorem Th20: :: CFUNCT_1:20
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds dom ((b2 ^ ) ^ ) = dom (b2 | (dom (b2 ^ )))
proof end;

theorem Th21: :: CFUNCT_1:21
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX
for b3 being Element of COMPLEX holds
( b3 <> 0 implies (b3 (#) b2) " {0} = b2 " {0} )
proof end;

theorem Th22: :: CFUNCT_1:22
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds (b2 + b3) + b4 = b2 + (b3 + b4)
proof end;

theorem Th23: :: CFUNCT_1:23
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds (b2 (#) b3) (#) b4 = b2 (#) (b3 (#) b4)
proof end;

theorem Th24: :: CFUNCT_1:24
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds (b2 + b3) (#) b4 = (b2 (#) b4) + (b3 (#) b4)
proof end;

theorem Th25: :: CFUNCT_1:25
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds b2 (#) (b3 + b4) = (b2 (#) b3) + (b2 (#) b4) by Th24;

theorem Th26: :: CFUNCT_1:26
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX
for b4 being Element of COMPLEX holds b4 (#) (b2 (#) b3) = (b4 (#) b2) (#) b3
proof end;

theorem Th27: :: CFUNCT_1:27
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX
for b4 being Element of COMPLEX holds b4 (#) (b2 (#) b3) = b2 (#) (b4 (#) b3)
proof end;

theorem Th28: :: CFUNCT_1:28
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds (b2 - b3) (#) b4 = (b2 (#) b4) - (b3 (#) b4)
proof end;

theorem Th29: :: CFUNCT_1:29
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds (b2 (#) b3) - (b2 (#) b4) = b2 (#) (b3 - b4) by Th28;

theorem Th30: :: CFUNCT_1:30
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX
for b4 being Element of COMPLEX holds b4 (#) (b2 + b3) = (b4 (#) b2) + (b4 (#) b3)
proof end;

theorem Th31: :: CFUNCT_1:31
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX
for b3, b4 being Element of COMPLEX holds (b3 * b4) (#) b2 = b3 (#) (b4 (#) b2)
proof end;

theorem Th32: :: CFUNCT_1:32
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX
for b4 being Element of COMPLEX holds b4 (#) (b2 - b3) = (b4 (#) b2) - (b4 (#) b3)
proof end;

theorem Th33: :: CFUNCT_1:33
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds b2 - b3 = (- 1r ) (#) (b3 - b2)
proof end;

theorem Th34: :: CFUNCT_1:34
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds b2 - (b3 + b4) = (b2 - b3) - b4
proof end;

theorem Th35: :: CFUNCT_1:35
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds 1r (#) b2 = b2
proof end;

theorem Th36: :: CFUNCT_1:36
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds b2 - (b3 - b4) = (b2 - b3) + b4
proof end;

theorem Th37: :: CFUNCT_1:37
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds b2 + (b3 - b4) = (b2 + b3) - b4
proof end;

theorem Th38: :: CFUNCT_1:38
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds |.(b2 (#) b3).| = |.b2.| (#) |.b3.|
proof end;

theorem Th39: :: CFUNCT_1:39
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX
for b3 being Element of COMPLEX holds |.(b3 (#) b2).| = |.b3.| (#) |.b2.|
proof end;

theorem Th40: :: CFUNCT_1:40
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds - b2 = (- 1r ) (#) b2
proof end;

theorem Th41: :: CFUNCT_1:41
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds - (- b2) = b2
proof end;

theorem Th42: :: CFUNCT_1:42
canceled;

theorem Th43: :: CFUNCT_1:43
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds b2 - (- b3) = b2 + b3
proof end;

theorem Th44: :: CFUNCT_1:44
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds (b2 ^ ) ^ = b2 | (dom (b2 ^ ))
proof end;

theorem Th45: :: CFUNCT_1:45
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds (b2 (#) b3) ^ = (b2 ^ ) (#) (b3 ^ )
proof end;

theorem Th46: :: CFUNCT_1:46
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX
for b3 being Element of COMPLEX holds
( b3 <> 0 implies (b3 (#) b2) ^ = (b3 " ) (#) (b2 ^ ) )
proof end;

Lemma30: (- 1r ) " = - 1r
by COMPLEX1:def 7;

theorem Th47: :: CFUNCT_1:47
canceled;

theorem Th48: :: CFUNCT_1:48
canceled;

theorem Th49: :: CFUNCT_1:49
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds (- b2) ^ = (- 1r ) (#) (b2 ^ )
proof end;

theorem Th50: :: CFUNCT_1:50
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds |.b2.| ^ = |.(b2 ^ ).|
proof end;

theorem Th51: :: CFUNCT_1:51
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds b2 / b3 = b2 (#) (b3 ^ )
proof end;

theorem Th52: :: CFUNCT_1:52
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX
for b4 being Element of COMPLEX holds b4 (#) (b2 / b3) = (b4 (#) b2) / b3
proof end;

theorem Th53: :: CFUNCT_1:53
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds (b2 / b3) (#) b3 = b2 | (dom (b3 ^ ))
proof end;

theorem Th54: :: CFUNCT_1:54
for b1 being non empty set
for b2, b3, b4, b5 being PartFunc of b1, COMPLEX holds (b2 / b3) (#) (b4 / b5) = (b2 (#) b4) / (b3 (#) b5)
proof end;

theorem Th55: :: CFUNCT_1:55
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds (b2 / b3) ^ = (b3 | (dom (b3 ^ ))) / b2
proof end;

theorem Th56: :: CFUNCT_1:56
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds b2 (#) (b3 / b4) = (b2 (#) b3) / b4
proof end;

theorem Th57: :: CFUNCT_1:57
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds b2 / (b3 / b4) = (b2 (#) (b4 | (dom (b4 ^ )))) / b3
proof end;

theorem Th58: :: CFUNCT_1:58
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds
( - (b2 / b3) = (- b2) / b3 & b2 / (- b3) = - (b2 / b3) )
proof end;

theorem Th59: :: CFUNCT_1:59
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, COMPLEX holds
( (b2 / b3) + (b4 / b3) = (b2 + b4) / b3 & (b2 / b3) - (b4 / b3) = (b2 - b4) / b3 )
proof end;

theorem Th60: :: CFUNCT_1:60
for b1 being non empty set
for b2, b3, b4, b5 being PartFunc of b1, COMPLEX holds (b2 / b3) + (b4 / b5) = ((b2 (#) b5) + (b4 (#) b3)) / (b3 (#) b5)
proof end;

theorem Th61: :: CFUNCT_1:61
for b1 being non empty set
for b2, b3, b4, b5 being PartFunc of b1, COMPLEX holds (b2 / b3) / (b4 / b5) = (b2 (#) (b5 | (dom (b5 ^ )))) / (b3 (#) b4)
proof end;

theorem Th62: :: CFUNCT_1:62
for b1 being non empty set
for b2, b3, b4, b5 being PartFunc of b1, COMPLEX holds (b2 / b3) - (b4 / b5) = ((b2 (#) b5) - (b4 (#) b3)) / (b3 (#) b5)
proof end;

theorem Th63: :: CFUNCT_1:63
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds |.(b2 / b3).| = |.b2.| / |.b3.|
proof end;

theorem Th64: :: CFUNCT_1:64
for b1 being set
for b2 being non empty set
for b3, b4 being PartFunc of b2, COMPLEX holds
( (b3 + b4) | b1 = (b3 | b1) + (b4 | b1) & (b3 + b4) | b1 = (b3 | b1) + b4 & (b3 + b4) | b1 = b3 + (b4 | b1) )
proof end;

theorem Th65: :: CFUNCT_1:65
for b1 being set
for b2 being non empty set
for b3, b4 being PartFunc of b2, COMPLEX holds
( (b3 (#) b4) | b1 = (b3 | b1) (#) (b4 | b1) & (b3 (#) b4) | b1 = (b3 | b1) (#) b4 & (b3 (#) b4) | b1 = b3 (#) (b4 | b1) )
proof end;

theorem Th66: :: CFUNCT_1:66
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX holds
( (- b3) | b1 = - (b3 | b1) & (b3 ^ ) | b1 = (b3 | b1) ^ & |.b3.| | b1 = |.(b3 | b1).| )
proof end;

theorem Th67: :: CFUNCT_1:67
for b1 being set
for b2 being non empty set
for b3, b4 being PartFunc of b2, COMPLEX holds
( (b3 - b4) | b1 = (b3 | b1) - (b4 | b1) & (b3 - b4) | b1 = (b3 | b1) - b4 & (b3 - b4) | b1 = b3 - (b4 | b1) )
proof end;

theorem Th68: :: CFUNCT_1:68
for b1 being set
for b2 being non empty set
for b3, b4 being PartFunc of b2, COMPLEX holds
( (b3 / b4) | b1 = (b3 | b1) / (b4 | b1) & (b3 / b4) | b1 = (b3 | b1) / b4 & (b3 / b4) | b1 = b3 / (b4 | b1) )
proof end;

theorem Th69: :: CFUNCT_1:69
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX
for b4 being Element of COMPLEX holds (b4 (#) b3) | b1 = b4 (#) (b3 | b1)
proof end;

theorem Th70: :: CFUNCT_1:70
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds
( ( b2 is total & b3 is total implies b2 + b3 is total ) & ( b2 + b3 is total implies ( b2 is total & b3 is total ) ) & ( b2 is total & b3 is total implies b2 - b3 is total ) & ( b2 - b3 is total implies ( b2 is total & b3 is total ) ) & ( b2 is total & b3 is total implies b2 (#) b3 is total ) & ( b2 (#) b3 is total implies ( b2 is total & b3 is total ) ) )
proof end;

theorem Th71: :: CFUNCT_1:71
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX
for b3 being Element of COMPLEX holds
( b2 is total iff b3 (#) b2 is total )
proof end;

theorem Th72: :: CFUNCT_1:72
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds
( b2 is total iff - b2 is total )
proof end;

theorem Th73: :: CFUNCT_1:73
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds
( b2 is total iff |.b2.| is total )
proof end;

theorem Th74: :: CFUNCT_1:74
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX holds
( b2 ^ is total iff ( b2 " {0} = {} & b2 is total ) )
proof end;

theorem Th75: :: CFUNCT_1:75
for b1 being non empty set
for b2, b3 being PartFunc of b1, COMPLEX holds
( ( b2 is total & b3 " {0} = {} & b3 is total ) iff b2 / b3 is total )
proof end;

theorem Th76: :: CFUNCT_1:76
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being PartFunc of b1, COMPLEX holds
( b3 is total & b4 is total implies ( (b3 + b4) /. b2 = (b3 /. b2) + (b4 /. b2) & (b3 - b4) /. b2 = (b3 /. b2) - (b4 /. b2) & (b3 (#) b4) /. b2 = (b3 /. b2) * (b4 /. b2) ) )
proof end;

theorem Th77: :: CFUNCT_1:77
for b1 being non empty set
for b2 being Element of b1
for b3 being PartFunc of b1, COMPLEX
for b4 being Element of COMPLEX holds
( b3 is total implies (b4 (#) b3) /. b2 = b4 * (b3 /. b2) )
proof end;

theorem Th78: :: CFUNCT_1:78
for b1 being non empty set
for b2 being Element of b1
for b3 being PartFunc of b1, COMPLEX holds
( b3 is total implies ( (- b3) /. b2 = - (b3 /. b2) & |.b3.| . b2 = |.(b3 /. b2).| ) )
proof end;

theorem Th79: :: CFUNCT_1:79
for b1 being non empty set
for b2 being Element of b1
for b3 being PartFunc of b1, COMPLEX holds
( b3 ^ is total implies (b3 ^ ) /. b2 = (b3 /. b2) " )
proof end;

theorem Th80: :: CFUNCT_1:80
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being PartFunc of b1, COMPLEX holds
( b3 is total & b4 ^ is total implies (b3 / b4) /. b2 = (b3 /. b2) * ((b4 /. b2) " ) )
proof end;

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, COMPLEX ;
let c3 be set ;
pred c2 is_bounded_on c3 means :Def3: :: CFUNCT_1:def 3
|.a2.| is_bounded_on a3;
end;

:: deftheorem Def3 defines is_bounded_on CFUNCT_1:def 3 :
for b1 being non empty set
for b2 being PartFunc of b1, COMPLEX
for b3 being set holds
( b2 is_bounded_on b3 iff |.b2.| is_bounded_on b3 );

theorem Th81: :: CFUNCT_1:81
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX holds
( b3 is_bounded_on b1 iff ex b4 being real number st
for b5 being Element of b2 holds
( b5 in b1 /\ (dom b3) implies |.(b3 /. b5).| <= b4 ) )
proof end;

theorem Th82: :: CFUNCT_1:82
for b1, b2 being set
for b3 being non empty set
for b4 being PartFunc of b3, COMPLEX holds
( b1 c= b2 & b4 is_bounded_on b2 implies b4 is_bounded_on b1 )
proof end;

theorem Th83: :: CFUNCT_1:83
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX holds
( b1 misses dom b3 implies b3 is_bounded_on b1 )
proof end;

theorem Th84: :: CFUNCT_1:84
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX
for b4 being Element of COMPLEX holds
( b3 is_bounded_on b1 implies b4 (#) b3 is_bounded_on b1 )
proof end;

theorem Th85: :: CFUNCT_1:85
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX holds |.b3.| is_bounded_below_on b1
proof end;

theorem Th86: :: CFUNCT_1:86
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX holds
( b3 is_bounded_on b1 implies ( |.b3.| is_bounded_on b1 & - b3 is_bounded_on b1 ) )
proof end;

theorem Th87: :: CFUNCT_1:87
for b1, b2 being set
for b3 being non empty set
for b4, b5 being PartFunc of b3, COMPLEX holds
( b4 is_bounded_on b1 & b5 is_bounded_on b2 implies b4 + b5 is_bounded_on b1 /\ b2 )
proof end;

theorem Th88: :: CFUNCT_1:88
for b1, b2 being set
for b3 being non empty set
for b4, b5 being PartFunc of b3, COMPLEX holds
( b4 is_bounded_on b1 & b5 is_bounded_on b2 implies ( b4 (#) b5 is_bounded_on b1 /\ b2 & b4 - b5 is_bounded_on b1 /\ b2 ) )
proof end;

theorem Th89: :: CFUNCT_1:89
for b1, b2 being set
for b3 being non empty set
for b4 being PartFunc of b3, COMPLEX holds
( b4 is_bounded_on b1 & b4 is_bounded_on b2 implies b4 is_bounded_on b1 \/ b2 )
proof end;

theorem Th90: :: CFUNCT_1:90
for b1, b2 being set
for b3 being non empty set
for b4, b5 being PartFunc of b3, COMPLEX holds
( b4 is_constant_on b1 & b5 is_constant_on b2 implies ( b4 + b5 is_constant_on b1 /\ b2 & b4 - b5 is_constant_on b1 /\ b2 & b4 (#) b5 is_constant_on b1 /\ b2 ) )
proof end;

theorem Th91: :: CFUNCT_1:91
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX
for b4 being Element of COMPLEX holds
( b3 is_constant_on b1 implies b4 (#) b3 is_constant_on b1 )
proof end;

theorem Th92: :: CFUNCT_1:92
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX holds
( b3 is_constant_on b1 implies ( |.b3.| is_constant_on b1 & - b3 is_constant_on b1 ) )
proof end;

theorem Th93: :: CFUNCT_1:93
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX holds
( b3 is_constant_on b1 implies b3 is_bounded_on b1 )
proof end;

theorem Th94: :: CFUNCT_1:94
for b1 being set
for b2 being non empty set
for b3 being PartFunc of b2, COMPLEX holds
( b3 is_constant_on b1 implies ( ( for b4 being Element of COMPLEX holds b4 (#) b3 is_bounded_on b1 ) & - b3 is_bounded_on b1 & |.b3.| is_bounded_on b1 ) )
proof end;

theorem Th95: :: CFUNCT_1:95
for b1, b2 being set
for b3 being non empty set
for b4, b5 being PartFunc of b3, COMPLEX holds
( b4 is_bounded_on b1 & b5 is_constant_on b2 implies b4 + b5 is_bounded_on b1 /\ b2 )
proof end;

theorem Th96: :: CFUNCT_1:96
for b1, b2 being set
for b3 being non empty set
for b4, b5 being PartFunc of b3, COMPLEX holds
( b4 is_bounded_on b1 & b5 is_constant_on b2 implies ( b4 - b5 is_bounded_on b1 /\ b2 & b5 - b4 is_bounded_on b1 /\ b2 & b4 (#) b5 is_bounded_on b1 /\ b2 ) )
proof end;