let X1, X2 be non empty set ; for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let S1 be SigmaField of X1; for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let S2 be SigmaField of X2; for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let M1 be sigma_Measure of S1; for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let M2 be sigma_Measure of S2; for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let E be Element of sigma (measurable_rectangles (S1,S2)); for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let A be Element of S1; for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let B be Element of S2; for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let x be Element of X1; for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let y be Element of X2; ( E = [:A,B:] implies ( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) ) )
assume A1:
E = [:A,B:]
; ( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
then A2:
Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . B) * ((chi (A,X1)) . x)
by Th47;
A3:
(M2 . B) * ((chi (A,X1)) . x) = M2 . (Measurable-X-section (E,x))
by A1, Th48;
A4:
Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . A) * ((chi (B,X2)) . y)
by A1, Th49;
A5:
(M1 . A) * ((chi (B,X2)) . y) = M1 . (Measurable-Y-section (E,y))
by A1, Th50;
thus
Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x)
Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y)