
theorem Th29:
for X1,X2 be non empty set, x be Element of X1, y be Element of X2,
 r be Real, f be PartFunc of [:X1,X2:],ExtREAL holds
  ProjPMap1(r(#)f,x) = r(#)ProjPMap1(f,x) &
  ProjPMap2(r(#)f,y) = r(#)ProjPMap2(f,y)
proof
    let X1,X2 be non empty set, x be Element of X1, y be Element of X2,
    r be Real, f be PartFunc of [:X1,X2:],ExtREAL;
    dom(ProjPMap1(r(#)f,x)) = X-section(dom(r(#)f),x)
  & dom(ProjPMap2(r(#)f,y)) = Y-section(dom(r(#)f),y)
      by Def3,Def4; then
A1: dom(ProjPMap1(r(#)f,x)) = X-section(dom f,x)
  & dom(ProjPMap2(r(#)f,y)) = Y-section(dom f,y) by MESFUNC1:def 6;
    dom(r(#)ProjPMap1(f,x)) = dom(ProjPMap1(f,x))
  & dom(r(#)ProjPMap2(f,y)) = dom(ProjPMap2(f,y)) by MESFUNC1:def 6; then
A2: dom(r(#)ProjPMap1(f,x)) = X-section(dom f,x)
  & dom(r(#)ProjPMap2(f,y)) = Y-section(dom f,y) by Def3,Def4;

    now let y be Element of X2;
     assume A3: y in dom(ProjPMap1(r(#)f,x)); then
     y in {y where y is Element of X2: [x,y] in dom f}
       by A1,MEASUR11:def 4; then
A4:  ex y1 be Element of X2 st y1 = y & [x,y1] in dom f; then
A5:  [x,y] in dom(r(#)f) by MESFUNC1:def 6;

A6:  f.(x,y) = f.[x,y];
     (r(#)ProjPMap1(f,x)).y = r * ProjPMap1(f,x).y by A1,A2,A3,MESFUNC1:def 6
      .= r * (f.[x,y]) by A4,A6,Def3
      .= (r(#)f).(x,y) by A5,MESFUNC1:def 6;
     hence ProjPMap1(r(#)f,x).y = (r(#)ProjPMap1(f,x)).y by A5,Def3;
    end;
    hence ProjPMap1(r(#)f,x) = r(#)ProjPMap1(f,x) by A1,A2,PARTFUN1:5;

    now let x be Element of X1;
     assume A7: x in dom(ProjPMap2(r(#)f,y)); then
     x in {x where x is Element of X1: [x,y] in dom f}
       by A1,MEASUR11:def 5; then
A8:  ex x1 be Element of X1 st x1 = x & [x1,y] in dom f; then
A9:  [x,y] in dom(r(#)f) by MESFUNC1:def 6;

A10: f.(x,y) = f.[x,y];
     (r(#)ProjPMap2(f,y)).x = r * ProjPMap2(f,y).x by A1,A2,A7,MESFUNC1:def 6
      .= r * (f.[x,y]) by A8,A10,Def4
      .= (r(#)f).(x,y) by A9,MESFUNC1:def 6;
     hence ProjPMap2(r(#)f,y).x = (r(#)ProjPMap2(f,y)).x by A9,Def4;
    end;
    hence ProjPMap2(r(#)f,y) = r(#)ProjPMap2(f,y) by A1,A2,PARTFUN1:5;
end;
