let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
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 st E = [:A,B:] holds
M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
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 st E = [:A,B:] holds
M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)
let S2 be SigmaField of X2; :: thesis: 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 st E = [:A,B:] holds
M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)
let M2 be sigma_Measure of S2; :: thesis: 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 st E = [:A,B:] holds
M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for A being Element of S1
for B being Element of S2
for x being Element of X1 st E = [:A,B:] holds
M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)
let A be Element of S1; :: thesis: for B being Element of S2
for x being Element of X1 st E = [:A,B:] holds
M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)
let B be Element of S2; :: thesis: for x being Element of X1 st E = [:A,B:] holds
M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)
let x be Element of X1; :: thesis: ( E = [:A,B:] implies M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x) )
assume A1: E = [:A,B:] ; :: thesis: M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)

per cases ( x in A or not x in A ) ;

suppose A4: x in A ; :: thesis: M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)

then A2: M2 . (Measurable-X-section (E,x)) = M2 . B by A1, Th16;
(chi (A,X1)) . x = 1 by A4, FUNCT_3:def 3;
hence M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x) by A2, XXREAL_3:81; :: thesis: verum

end;

suppose A5: not x in A ; :: thesis: M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x)

then Measurable-X-section (E,x) = {} by A1, Th16;
then A3: M2 . (Measurable-X-section (E,x)) = 0 by VALUED_0:def 19;
(chi (A,X1)) . x = 0 by A5, FUNCT_3:def 3;
hence M2 . (Measurable-X-section (E,x)) = (M2 . B) * ((chi (A,X1)) . x) by A3; :: thesis: verum

end;

end;