
theorem Th25:
for X,Y,Z be non empty set, A be Subset of X, B be Subset of Y,
 x be Element of X, f be PartFunc of [:X,Y:],Z st dom f = [:A,B:] holds
  ( x in A implies dom ProjPMap1(f,x) = B ) &
  ( not x in A implies dom ProjPMap1(f,x) = {} )
proof
    let X,Y,Z be non empty set, A be Subset of X, B be Subset of Y,
    x be Element of X, f be PartFunc of [:X,Y:],Z;
    assume
A1:  dom f = [:A,B:];

    hereby assume
A2:  x in A;
     for y be Element of Y holds y in X-section(dom f,x) iff y in B
     proof
      let y be Element of Y;
      hereby assume y in X-section(dom f,x); then
       [x,y] in dom f by MESFUN12:25;
       hence y in B by A1,ZFMISC_1:87;
      end;
      assume y in B; then
      [x,y] in dom f by A1,A2,ZFMISC_1:87; then
      y in { y where y is Element of Y : [x,y] in dom f };
      hence y in X-section(dom f,x) by MEASUR11:def 4;
     end; then
     X-section(dom f,x) = B by SUBSET_1:3;
     hence dom ProjPMap1(f,x) = B by MESFUN12:def 3;
    end;
    assume A3: not x in A;
    now assume X-section(dom f,x) <> {}; then
     consider y be object such that
A4:  y in X-section(dom f,x) by XBOOLE_0:def 1;
     reconsider y as Element of Y by A4;
     [x,y] in dom f by A4,MESFUN12:25;
     hence contradiction by A1,A3,ZFMISC_1:87;
    end;
    hence dom ProjPMap1(f,x) = {} by MESFUN12:def 3;
end;
