reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;

theorem Th29:
  for f being Function st Z c= Y holds <:f,X,Z:> c= <:f,X,Y:>
proof
  let f be Function such that
A1: Z c= Y;
A2: dom <:f,X,Z:> c= dom <:f,X,Y:>
  proof
    let x be object;
    assume
A3: x in dom <:f,X,Z:>;
    then f.x in Z & x in dom f by Th24;
    hence thesis by A1,A3,Th24;
  end;
  now
    let x be object;
    assume
A4: x in dom <:f,X,Z:>;
    then <:f,X,Z:>.x = f.x by Th26;
    hence <:f,X,Z:>.x = <:f,X,Y:>.x by A2,A4,Th26;
  end;
  hence thesis by A2,GRFUNC_1:2;
end;
