
theorem Th47:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  f be PartFunc of [:X1,X2:],ExtREAL, x be Element of X1, y be Element of X2,
  E be Element of sigma measurable_rectangles(S1,S2)
 st E c= dom f & f is E-measurable
 holds ProjPMap1(f,x) is (Measurable-X-section(E,x))-measurable &
       ProjPMap2(f,y) is (Measurable-Y-section(E,y))-measurable
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
   f be PartFunc of [:X1,X2:],ExtREAL, x be Element of X1, y be Element of X2,
   A be Element of sigma measurable_rectangles(S1,S2);
   assume that
A1: A c= dom f and
A2: f is A-measurable;
   X-section(A,x) c= X-section(dom f,x)
 & Y-section(A,y) c= Y-section(dom f,y) by A1,MEASUR11:20,21; then
   Measurable-X-section(A,x) c= X-section(dom f,x)
 & Measurable-Y-section(A,y) c= Y-section(dom f,y)
   by MEASUR11:def 6,def 7; then
A3:Measurable-X-section(A,x) c= dom(ProjPMap1(f,x))
 & Measurable-Y-section(A,y) c= dom(ProjPMap2(f,y)) by Def3,Def4;
   ProjPMap1(max+f,x) is (Measurable-X-section(A,x))-measurable
 & ProjPMap2(max+f,y) is (Measurable-Y-section(A,y))-measurable
 & ProjPMap1(max-f,x) is (Measurable-X-section(A,x))-measurable
 & ProjPMap2(max-f,y) is (Measurable-Y-section(A,y))-measurable
    by A1,A2,Lm4; then
   max+(ProjPMap1(f,x)) is (Measurable-X-section(A,x))-measurable
 & max+(ProjPMap2(f,y)) is (Measurable-Y-section(A,y))-measurable
 & max-(ProjPMap1(f,x)) is (Measurable-X-section(A,x))-measurable
 & max-(ProjPMap2(f,y)) is (Measurable-Y-section(A,y))-measurable
    by Th45,Th46;
   hence thesis by A3,MESFUN11:10;
end;
