let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for y being Element of X2 st E = [:A,B:] holds
M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for y being Element of X2 st E = [:A,B:] holds
M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for y being Element of X2 st E = [:A,B:] holds
M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)
let M1 be sigma_Measure of S1; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for y being Element of X2 st E = [:A,B:] holds
M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for A being Element of S1
for B being Element of S2
for y being Element of X2 st E = [:A,B:] holds
M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)
let A be Element of S1; :: thesis: for B being Element of S2
for y being Element of X2 st E = [:A,B:] holds
M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)
let B be Element of S2; :: thesis: for y being Element of X2 st E = [:A,B:] holds
M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)
let y be Element of X2; :: thesis: ( E = [:A,B:] implies M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y) )
assume A1: E = [:A,B:] ; :: thesis: M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)

per cases ( y in B or not y in B ) ;

suppose A4: y in B ; :: thesis: M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)

then A2: M1 . (Measurable-Y-section (E,y)) = M1 . A by A1, Th16;
(chi (B,X2)) . y = 1 by A4, FUNCT_3:def 3;
hence M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y) by A2, XXREAL_3:81; :: thesis: verum

end;

suppose A5: not y in B ; :: thesis: M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y)

then Measurable-Y-section (E,y) = {} by A1, Th16;
then A3: M1 . (Measurable-Y-section (E,y)) = 0 by VALUED_0:def 19;
(chi (B,X2)) . y = 0 by A5, FUNCT_3:def 3;
hence M1 . (Measurable-Y-section (E,y)) = (M1 . A) * ((chi (B,X2)) . y) by A3; :: thesis: verum

end;

end;