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 Th31:
  for f being Function st dom f c= X & rng f c= Y holds f = <:f,X, Y:>
proof
  let f be Function such that
A1: dom f c= X & rng f c= Y;
A2: dom f c= dom <:f,X,Y:>
  proof
    let x be object;
    assume
A3: x in dom f;
    then f.x in rng f by FUNCT_1:def 3;
    hence thesis by A1,A3,Th24;
  end;
  dom <:f,X,Y:> c= dom f by Th23;
  then
A4: dom f = dom <:f,X,Y:> by A2;
  for x being object st x in dom f holds f.x = <:f,X,Y:>.x by A2,Th26;
  hence thesis by A4;
end;
