
theorem Th27:
for X,Y 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:],REAL st dom f = [:A,B:] holds
  ( x in A implies
      dom ProjPMap1(R_EAL f,x) = B & dom ProjPMap1(|.R_EAL f.|,x) = B ) &
  ( not x in A implies
      dom ProjPMap1(R_EAL f,x) = {} & dom ProjPMap1(|.R_EAL f.|,x) = {} )
proof
    let X,Y 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:],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 x in A implies
     dom ProjPMap1(R_EAL f,x) = B & dom ProjPMap1(|.R_EAL f.|,x) = B
       by A1,Th25;
    assume not x in A;
    hence dom ProjPMap1(R_EAL f,x) = {} & dom ProjPMap1(|.R_EAL f.|,x) = {}
      by A1,A2,Th25;
end;
