
theorem Th26:
for X,Y,Z be non empty set, A be Subset of X, B be Subset of Y,
 y be Element of Y, f be PartFunc of [:X,Y:],Z st dom f = [:A,B:] holds
  ( y in B implies dom ProjPMap2(f,y) = A ) &
  ( not y in B implies dom ProjPMap2(f,y) = {} )
proof
    let X,Y,Z be non empty set, A be Subset of X, B be Subset of Y,
    y be Element of Y, f be PartFunc of [:X,Y:],Z;
    assume
A1:  dom f = [:A,B:];
    hereby assume
A2:  y in B;
     for x be Element of X holds x in Y-section(dom f,y) iff x in A
     proof
      let x be Element of X;
      hereby assume x in Y-section(dom f,y); then
       [x,y] in dom f by MESFUN12:25;
       hence x in A by A1,ZFMISC_1:87;
      end;
      assume x in A; then
      [x,y] in dom f by A1,A2,ZFMISC_1:87; then
      x in { x where x is Element of X : [x,y] in dom f };
      hence x in Y-section(dom f,y) by MEASUR11:def 5;
     end; then
     Y-section(dom f,y) = A by SUBSET_1:3;
     hence dom ProjPMap2(f,y) = A by MESFUN12:def 4;
    end;
    assume A3: not y in B;
    now assume Y-section(dom f,y) <> {}; then
     consider x be object such that
A4:  x in Y-section(dom f,y) by XBOOLE_0:def 1;
     reconsider x as Element of X by A4;
     [x,y] in dom f by A4,MESFUN12:25;
     hence contradiction by A1,A3,ZFMISC_1:87;
    end;
    hence dom ProjPMap2(f,y) = {} by MESFUN12:def 4;
end;
