
theorem Th28:
for X,Y 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:],REAL st dom f = [:A,B:] holds
  ( y in B implies
      dom ProjPMap2(R_EAL f,y) = A & dom ProjPMap2(|.R_EAL f.|,y) = A ) &
  ( not y in B implies
      dom ProjPMap2(R_EAL f,y) = {} & dom ProjPMap2(|.R_EAL f.|,y) = {} )
proof
    let X,Y 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:],REAL;
    assume dom f = [:A,B:]; then
A1: dom (R_EAL f) = [:A,B:] by MESFUNC5:def 7; then
A2: dom |.R_EAL f.| = [:A,B:] by MESFUNC1:def 10;

    hence y in B implies
     dom ProjPMap2(R_EAL f,y) = A & dom ProjPMap2(|.R_EAL f.|,y) = A
       by A1,Th26;
    assume not y in B;
    hence dom ProjPMap2(R_EAL f,y) = {} & dom ProjPMap2(|.R_EAL f.|,y) = {}
      by A1,A2,Th26;
end;
