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 X c= dom f & rng f c= Y holds <:f,X,Y:> is total
proof
  let f be Function such that
A1: X c= dom f and
A2: rng f c= Y;
  X c= dom <:f,X,Y:>
  proof
    let x be object;
    assume
A3: x in X;
    then f.x in rng f by A1,FUNCT_1:def 3;
    hence thesis by A1,A2,A3,Th24;
  end;
  hence dom <:f,X,Y:> = X;
end;
