
theorem Th44:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
 E be Element of sigma(measurable_rectangles(S1,S2)) holds
  ( for p be set holds X-section(E,p) in S2 )
& ( for p be set holds Y-section(E,p) in S1 )
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
       E be Element of sigma(measurable_rectangles(S1,S2));
   set K = {C where C is Subset of [:X1,X2:] :
             for x be set holds X-section(C,x) in S2};
   reconsider K as SigmaField of [:X1,X2:] by Th42;
A1:measurable_rectangles(S1,S2)
     c= Field_generated_by measurable_rectangles(S1,S2)
 & Field_generated_by measurable_rectangles(S1,S2) c= K
     by Th42,SRINGS_3:21; then
   measurable_rectangles(S1,S2) c= K; then
   sigma(measurable_rectangles(S1,S2)) c= K by PROB_1:def 9; then
   E in K; then
   ex C be Subset of [:X1,X2:] st
    E = C & for x be set holds X-section(C,x) in S2;
   hence for x be set holds X-section(E,x) in S2;
   set K2 = {C where C is Subset of [:X1,X2:] :
             for x be set holds Y-section(C,x) in S1};
   reconsider K2 as SigmaField of [:X1,X2:] by Th43;
   Field_generated_by measurable_rectangles(S1,S2) c= K2 by Th43; then
   measurable_rectangles(S1,S2) c= K2 by A1; then
   sigma(measurable_rectangles(S1,S2)) c= K2 by PROB_1:def 9; then
   E in K2; then
   ex C be Subset of [:X1,X2:] st
    E = C & for x be set holds Y-section(C,x) in S1;
   hence for x be set holds Y-section(E,x) in S1;
end;
