:: INTEGRA4 semantic presentation

theorem Th1: :: INTEGRA4:1
for b1 being closed-interval Subset of REAL
for b2 being Element of divs b1 holds
( vol b1 = 0 implies len b2 = 1 )
proof end;

theorem Th2: :: INTEGRA4:2
for b1 being closed-interval Subset of REAL holds
( chi b1,b1 is_integrable_on b1 & integral (chi b1,b1) = vol b1 )
proof end;

theorem Th3: :: INTEGRA4:3
for b1 being closed-interval Subset of REAL
for b2 being PartFunc of b1, REAL
for b3 being Real holds
( ( b2 is total & rng b2 = {b3} ) iff b2 = b3 (#) (chi b1,b1) )
proof end;

theorem Th4: :: INTEGRA4:4
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL
for b3 being Real holds
( rng b2 = {b3} implies ( b2 is_integrable_on b1 & integral b2 = b3 * (vol b1) ) )
proof end;

theorem Th5: :: INTEGRA4:5
for b1 being closed-interval Subset of REAL
for b2 being Real holds
ex b3 being Function of b1, REAL st
( rng b3 = {b2} & b3 is_bounded_on b1 )
proof end;

Lemma6: for b1 being closed-interval Subset of REAL
for b2 being PartFunc of b1, REAL
for b3 being Element of divs b1 holds
( vol b1 = 0 implies ( b2 is_upper_integrable_on b1 & upper_integral b2 = 0 ) )
proof end;

Lemma7: for b1 being closed-interval Subset of REAL
for b2 being PartFunc of b1, REAL
for b3 being Element of divs b1 holds
( vol b1 = 0 implies ( b2 is_lower_integrable_on b1 & lower_integral b2 = 0 ) )
proof end;

theorem Th6: :: INTEGRA4:6
for b1 being closed-interval Subset of REAL
for b2 being PartFunc of b1, REAL
for b3 being Element of divs b1 holds
( vol b1 = 0 implies ( b2 is_integrable_on b1 & integral b2 = 0 ) )
proof end;

theorem Th7: :: INTEGRA4:7
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL holds
not ( b2 is_bounded_on b1 & b2 is_integrable_on b1 & ( for b3 being Real holds
not ( inf (rng b2) <= b3 & b3 <= sup (rng b2) & integral b2 = b3 * (vol b1) ) ) )
proof end;

theorem Th8: :: INTEGRA4:8
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL
for b3 being DivSequence of b1 holds
( b2 is_bounded_on b1 & delta b3 is convergent & lim (delta b3) = 0 implies ( lower_sum b2,b3 is convergent & lim (lower_sum b2,b3) = lower_integral b2 ) )
proof end;

theorem Th9: :: INTEGRA4:9
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL
for b3 being DivSequence of b1 holds
( b2 is_bounded_on b1 & delta b3 is convergent & lim (delta b3) = 0 implies ( upper_sum b2,b3 is convergent & lim (upper_sum b2,b3) = upper_integral b2 ) )
proof end;

theorem Th10: :: INTEGRA4:10
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL holds
( b2 is_bounded_on b1 implies ( b2 is_upper_integrable_on b1 & b2 is_lower_integrable_on b1 ) )
proof end;

definition
let c1 be closed-interval Subset of REAL ;
let c2 be Element of divs c1;
let c3 be Nat;
pred c2 divide_into_equal c3 means :Def1: :: INTEGRA4:def 1
( len a2 = a3 & ( for b1 being Nat holds
( b1 in dom a2 implies a2 . b1 = (inf a1) + (((vol a1) / (len a2)) * b1) ) ) );
end;

:: deftheorem Def1 defines divide_into_equal INTEGRA4:def 1 :
for b1 being closed-interval Subset of REAL
for b2 being Element of divs b1
for b3 being Nat holds
( b2 divide_into_equal b3 iff ( len b2 = b3 & ( for b4 being Nat holds
( b4 in dom b2 implies b2 . b4 = (inf b1) + (((vol b1) / (len b2)) * b4) ) ) ) );

Lemma13: for b1 being closed-interval Subset of REAL
for b2 being Nat holds
not ( b2 > 0 & vol b1 > 0 & ( for b3 being Element of divs b1 holds
not ( len b3 = b2 & ( for b4 being Nat holds
( b4 in dom b3 implies b3 . b4 = (inf b1) + (((vol b1) / b2) * b4) ) ) ) ) )
proof end;

Lemma14: for b1 being closed-interval Subset of REAL holds
not ( vol b1 > 0 & ( for b2 being DivSequence of b1 holds
not ( delta b2 is convergent & lim (delta b2) = 0 ) ) )
proof end;

Lemma15: for b1 being closed-interval Subset of REAL holds
not ( vol b1 = 0 & ( for b2 being DivSequence of b1 holds
not ( delta b2 is convergent & lim (delta b2) = 0 ) ) )
proof end;

theorem Th11: :: INTEGRA4:11
for b1 being closed-interval Subset of REAL holds
ex b2 being DivSequence of b1 st
( delta b2 is convergent & lim (delta b2) = 0 )
proof end;

theorem Th12: :: INTEGRA4:12
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL holds
( b2 is_bounded_on b1 implies ( b2 is_integrable_on b1 iff for b3 being DivSequence of b1 holds
( delta b3 is convergent & lim (delta b3) = 0 implies (lim (upper_sum b2,b3)) - (lim (lower_sum b2,b3)) = 0 ) ) )
proof end;

theorem Th13: :: INTEGRA4:13
for b1 being non empty set
for b2 being Function of b1, REAL holds
( max+ b2 is total & max- b2 is total )
proof end;

theorem Th14: :: INTEGRA4:14
for b1 being non empty set
for b2 being set
for b3 being PartFunc of b1, REAL holds
( b3 is_bounded_above_on b2 implies max+ b3 is_bounded_above_on b2 )
proof end;

theorem Th15: :: INTEGRA4:15
for b1 being non empty set
for b2 being set
for b3 being PartFunc of b1, REAL holds max+ b3 is_bounded_below_on b2
proof end;

theorem Th16: :: INTEGRA4:16
for b1 being non empty set
for b2 being set
for b3 being PartFunc of b1, REAL holds
( b3 is_bounded_below_on b2 implies max- b3 is_bounded_above_on b2 )
proof end;

theorem Th17: :: INTEGRA4:17
for b1 being non empty set
for b2 being set
for b3 being PartFunc of b1, REAL holds max- b3 is_bounded_below_on b2
proof end;

theorem Th18: :: INTEGRA4:18
for b1 being closed-interval Subset of REAL
for b2 being set
for b3 being PartFunc of b1, REAL holds
( b3 is_bounded_above_on b1 implies rng (b3 | b2) is bounded_above )
proof end;

theorem Th19: :: INTEGRA4:19
for b1 being closed-interval Subset of REAL
for b2 being set
for b3 being PartFunc of b1, REAL holds
( b3 is_bounded_below_on b1 implies rng (b3 | b2) is bounded_below )
proof end;

theorem Th20: :: INTEGRA4:20
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL holds
( b2 is_bounded_on b1 & b2 is_integrable_on b1 implies max+ b2 is_integrable_on b1 )
proof end;

theorem Th21: :: INTEGRA4:21
for b1 being non empty set
for b2 being PartFunc of b1, REAL holds max- b2 = max+ (- b2)
proof end;

theorem Th22: :: INTEGRA4:22
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL holds
( b2 is_bounded_on b1 & b2 is_integrable_on b1 implies max- b2 is_integrable_on b1 )
proof end;

theorem Th23: :: INTEGRA4:23
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL holds
( b2 is_bounded_on b1 & b2 is_integrable_on b1 implies ( abs b2 is_integrable_on b1 & abs (integral b2) <= integral (abs b2) ) )
proof end;

theorem Th24: :: INTEGRA4:24
for b1 being Real
for b2 being closed-interval Subset of REAL
for b3 being Function of b2, REAL holds
( ( for b4, b5 being Real holds
( b4 in b2 & b5 in b2 implies abs ((b3 . b4) - (b3 . b5)) <= b1 ) ) implies (sup (rng b3)) - (inf (rng b3)) <= b1 )
proof end;

theorem Th25: :: INTEGRA4:25
for b1 being Real
for b2 being closed-interval Subset of REAL
for b3, b4 being Function of b2, REAL holds
( b3 is_bounded_on b2 & b1 >= 0 & ( for b5, b6 being Real holds
( b5 in b2 & b6 in b2 implies abs ((b4 . b5) - (b4 . b6)) <= b1 * (abs ((b3 . b5) - (b3 . b6))) ) ) implies (sup (rng b4)) - (inf (rng b4)) <= b1 * ((sup (rng b3)) - (inf (rng b3))) )
proof end;

theorem Th26: :: INTEGRA4:26
for b1 being Real
for b2 being closed-interval Subset of REAL
for b3, b4, b5 being Function of b2, REAL holds
( b3 is_bounded_on b2 & b4 is_bounded_on b2 & b1 >= 0 & ( for b6, b7 being Real holds
( b6 in b2 & b7 in b2 implies abs ((b5 . b6) - (b5 . b7)) <= b1 * ((abs ((b3 . b6) - (b3 . b7))) + (abs ((b4 . b6) - (b4 . b7)))) ) ) implies (sup (rng b5)) - (inf (rng b5)) <= b1 * (((sup (rng b3)) - (inf (rng b3))) + ((sup (rng b4)) - (inf (rng b4)))) )
proof end;

theorem Th27: :: INTEGRA4:27
for b1 being Real
for b2 being closed-interval Subset of REAL
for b3, b4 being Function of b2, REAL holds
( b3 is_bounded_on b2 & b3 is_integrable_on b2 & b4 is_bounded_on b2 & b1 > 0 & ( for b5, b6 being Real holds
( b5 in b2 & b6 in b2 implies abs ((b4 . b5) - (b4 . b6)) <= b1 * (abs ((b3 . b5) - (b3 . b6))) ) ) implies b4 is_integrable_on b2 )
proof end;

theorem Th28: :: INTEGRA4:28
for b1 being Real
for b2 being closed-interval Subset of REAL
for b3, b4, b5 being Function of b2, REAL holds
( b3 is_bounded_on b2 & b3 is_integrable_on b2 & b4 is_bounded_on b2 & b4 is_integrable_on b2 & b5 is_bounded_on b2 & b1 > 0 & ( for b6, b7 being Real holds
( b6 in b2 & b7 in b2 implies abs ((b5 . b6) - (b5 . b7)) <= b1 * ((abs ((b3 . b6) - (b3 . b7))) + (abs ((b4 . b6) - (b4 . b7)))) ) ) implies b5 is_integrable_on b2 )
proof end;

theorem Th29: :: INTEGRA4:29
for b1 being closed-interval Subset of REAL
for b2, b3 being Function of b1, REAL holds
( b2 is_bounded_on b1 & b2 is_integrable_on b1 & b3 is_bounded_on b1 & b3 is_integrable_on b1 implies b2 (#) b3 is_integrable_on b1 )
proof end;

theorem Th30: :: INTEGRA4:30
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL holds
( b2 is_bounded_on b1 & b2 is_integrable_on b1 & not 0 in rng b2 & b2 ^ is_bounded_on b1 implies b2 ^ is_integrable_on b1 )
proof end;