:: UNIFORM1 semantic presentation

theorem Th1: :: UNIFORM1:1
canceled;

theorem Th2: :: UNIFORM1:2
for b1 being Real holds
not ( b1 > 0 & ( for b2 being Nat holds
not ( b2 > 0 & 1 / b2 < b1 ) ) )
proof end;

definition
let c1, c2 be non empty MetrStruct ;
let c3 be Function of c1,c2;
attr a3 is uniformly_continuous means :Def1: :: UNIFORM1:def 1
for b1 being Real holds
not ( 0 < b1 & ( for b2 being Real holds
not ( 0 < b2 & ( for b3, b4 being Element of a1 holds
not ( dist b3,b4 < b2 & not dist (a3 /. b3),(a3 /. b4) < b1 ) ) ) ) );
end;

:: deftheorem Def1 defines uniformly_continuous UNIFORM1:def 1 :
for b1, b2 being non empty MetrStruct
for b3 being Function of b1,b2 holds
( b3 is uniformly_continuous iff for b4 being Real holds
not ( 0 < b4 & ( for b5 being Real holds
not ( 0 < b5 & ( for b6, b7 being Element of b1 holds
not ( dist b6,b7 < b5 & not dist (b3 /. b6),(b3 /. b7) < b4 ) ) ) ) ) );

theorem Th3: :: UNIFORM1:3
for b1 being non empty TopSpace
for b2 being non empty MetrSpace
for b3 being Function of b1,(TopSpaceMetr b2) holds
( b3 is continuous implies for b4 being Real
for b5 being Element of the carrier of b2
for b6 being Subset of (TopSpaceMetr b2) holds
( b6 = Ball b5,b4 implies b3 " b6 is open ) )
proof end;

theorem Th4: :: UNIFORM1:4
for b1, b2 being non empty MetrSpace
for b3 being Function of (TopSpaceMetr b1),(TopSpaceMetr b2) holds
( ( for b4 being real number
for b5 being Element of the carrier of b1
for b6 being Element of b2 holds
not ( b4 > 0 & b6 = b3 . b5 & ( for b7 being real number holds
not ( b7 > 0 & ( for b8 being Element of b1
for b9 being Element of b2 holds
not ( b9 = b3 . b8 & dist b5,b8 < b7 & not dist b6,b9 < b4 ) ) ) ) ) ) implies b3 is continuous )
proof end;

theorem Th5: :: UNIFORM1:5
for b1, b2 being non empty MetrSpace
for b3 being Function of (TopSpaceMetr b1),(TopSpaceMetr b2) holds
( b3 is continuous implies for b4 being Real
for b5 being Element of the carrier of b1
for b6 being Element of b2 holds
not ( b4 > 0 & b6 = b3 . b5 & ( for b7 being Real holds
not ( b7 > 0 & ( for b8 being Element of b1
for b9 being Element of b2 holds
not ( b9 = b3 . b8 & dist b5,b8 < b7 & not dist b6,b9 < b4 ) ) ) ) ) )
proof end;

theorem Th6: :: UNIFORM1:6
for b1, b2 being non empty MetrSpace
for b3 being Function of b1,b2
for b4 being Function of (TopSpaceMetr b1),(TopSpaceMetr b2) holds
( b3 = b4 & b3 is uniformly_continuous implies b4 is continuous )
proof end;

theorem Th7: :: UNIFORM1:7
for b1 being non empty MetrSpace
for b2 being Subset-Family of (TopSpaceMetr b1) holds
not ( b2 is_a_cover_of TopSpaceMetr b1 & b2 is open & TopSpaceMetr b1 is compact & ( for b3 being Real holds
not ( b3 > 0 & ( for b4, b5 being Element of b1 holds
not ( dist b4,b5 < b3 & ( for b6 being Subset of (TopSpaceMetr b1) holds
not ( b4 in b6 & b5 in b6 & b6 in b2 ) ) ) ) ) ) )
proof end;

theorem Th8: :: UNIFORM1:8
for b1, b2 being non empty MetrSpace
for b3 being Function of b1,b2
for b4 being Function of (TopSpaceMetr b1),(TopSpaceMetr b2) holds
( b4 = b3 & TopSpaceMetr b1 is compact & b4 is continuous implies b3 is uniformly_continuous )
proof end;

Lemma8: Closed-Interval-TSpace 0,1 = TopSpaceMetr (Closed-Interval-MSpace 0,1)
by TOPMETR:def 8;

Lemma9: I[01] = TopSpaceMetr (Closed-Interval-MSpace 0,1)
by TOPMETR:27, TOPMETR:def 8;

Lemma10: the carrier of I[01] = the carrier of (Closed-Interval-MSpace 0,1)
by Lemma8, TOPMETR:16, TOPMETR:27;

theorem Th9: :: UNIFORM1:9
for b1 being Nat
for b2 being Function of I[01] ,(TOP-REAL b1)
for b3 being Function of (Closed-Interval-MSpace 0,1),(Euclid b1) holds
( b2 is continuous & b3 = b2 implies b3 is uniformly_continuous ) by Lemma8, Th8, TOPMETR:27;

theorem Th10: :: UNIFORM1:10
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3 being non empty Subset of (Euclid b1)
for b4 being Function of I[01] ,((TOP-REAL b1) | b2)
for b5 being Function of (Closed-Interval-MSpace 0,1),((Euclid b1) | b3) holds
( b2 = b3 & b4 is continuous & b5 = b4 implies b5 is uniformly_continuous )
proof end;

theorem Th11: :: UNIFORM1:11
for b1 being Nat
for b2 being Function of I[01] ,(TOP-REAL b1) holds
ex b3 being Function of (Closed-Interval-MSpace 0,1),(Euclid b1) st b3 = b2
proof end;

Lemma13: for b1 being set
for b2 being FinSequence holds
( len (b2 ^ <*b1*>) = (len b2) + 1 & len (<*b1*> ^ b2) = (len b2) + 1 & (b2 ^ <*b1*>) . ((len b2) + 1) = b1 & (<*b1*> ^ b2) . 1 = b1 )
proof end;

Lemma14: for b1 being set
for b2 being FinSequence holds
( 1 <= len b2 implies ( (b2 ^ <*b1*>) . 1 = b2 . 1 & (<*b1*> ^ b2) . ((len b2) + 1) = b2 . (len b2) ) )
proof end;

theorem Th12: :: UNIFORM1:12
canceled;

theorem Th13: :: UNIFORM1:13
for b1, b2 being Real holds abs (b1 - b2) = abs (b2 - b1)
proof end;

Lemma16: for b1, b2, b3 being Real holds
( b1 in [.b2,b3.] iff ( b2 <= b1 & b1 <= b3 ) )
proof end;

theorem Th14: :: UNIFORM1:14
for b1, b2, b3, b4 being Real holds
( b1 in [.b3,b4.] & b2 in [.b3,b4.] implies abs (b1 - b2) <= b4 - b3 )
proof end;

definition
let c1 be FinSequence of REAL ;
attr a1 is decreasing means :Def2: :: UNIFORM1:def 2
for b1, b2 being Nat holds
not ( b1 in dom a1 & b2 in dom a1 & b1 < b2 & not a1 . b1 > a1 . b2 );
end;

:: deftheorem Def2 defines decreasing UNIFORM1:def 2 :
for b1 being FinSequence of REAL holds
( b1 is decreasing iff for b2, b3 being Nat holds
not ( b2 in dom b1 & b3 in dom b1 & b2 < b3 & not b1 . b2 > b1 . b3 ) );

Lemma19: for b1 being FinSequence of REAL holds
( ( for b2 being Nat holds
not ( 1 <= b2 & b2 < len b1 & not b1 /. b2 < b1 /. (b2 + 1) ) ) implies b1 is increasing )
proof end;

Lemma20: for b1 being FinSequence of REAL holds
( ( for b2 being Nat holds
not ( 1 <= b2 & b2 < len b1 & not b1 /. b2 > b1 /. (b2 + 1) ) ) implies b1 is decreasing )
proof end;

theorem Th15: :: UNIFORM1:15
for b1 being Nat
for b2 being Real
for b3 being Function of I[01] ,(TOP-REAL b1)
for b4, b5 being Element of (TOP-REAL b1) holds
not ( b2 > 0 & b3 is continuous & b3 is one-to-one & b3 . 0 = b4 & b3 . 1 = b5 & ( for b6 being FinSequence of REAL holds
not ( b6 . 1 = 0 & b6 . (len b6) = 1 & 5 <= len b6 & rng b6 c= the carrier of I[01] & b6 is increasing & ( for b7 being Nat
for b8 being Subset of I[01]
for b9 being Subset of (Euclid b1) holds
not ( 1 <= b7 & b7 < len b6 & b8 = [.(b6 /. b7),(b6 /. (b7 + 1)).] & b9 = b3 .: b8 & not diameter b9 < b2 ) ) ) ) )
proof end;

theorem Th16: :: UNIFORM1:16
for b1 being Nat
for b2 being Real
for b3 being Function of I[01] ,(TOP-REAL b1)
for b4, b5 being Element of (TOP-REAL b1) holds
not ( b2 > 0 & b3 is continuous & b3 is one-to-one & b3 . 0 = b4 & b3 . 1 = b5 & ( for b6 being FinSequence of REAL holds
not ( b6 . 1 = 1 & b6 . (len b6) = 0 & 5 <= len b6 & rng b6 c= the carrier of I[01] & b6 is decreasing & ( for b7 being Nat
for b8 being Subset of I[01]
for b9 being Subset of (Euclid b1) holds
not ( 1 <= b7 & b7 < len b6 & b8 = [.(b6 /. (b7 + 1)),(b6 /. b7).] & b9 = b3 .: b8 & not diameter b9 < b2 ) ) ) ) )
proof end;