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
  for f being Function st <:f,X,Y:> is total holds f.:X c= Y
proof
  let f be Function such that
A1: dom <:f,X,Y:> = X;
  let y be object;
A2: rng <:f,X,Y:> c= Y by RELAT_1:def 19;
  assume y in f.:X;
  then consider x being object such that
  x in dom f and
A3: x in X & y = f.x by FUNCT_1:def 6;
  <:f,X,Y:>.x = y & <:f,X,Y:>.x in rng <:f,X,Y:> by A1,A3,Th26,FUNCT_1:def 3;
  hence thesis by A2;
end;
