reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem
  for a,b,c,d being object, g being Function st dom g = {a,b} & g.a = c & g
  .b = d holds g = (a,b) --> (c,d)
proof
  let a,b,c,d be object;
  let h be Function such that
A1: dom h = {a,b} and
A2: h.a = c and
A3: h.b = d;
  set f = {a} --> c, g = {b} -->d;
A4: b in {b} by TARSKI:def 1;
A5: a in {a} by TARSKI:def 1;
A6: now
    let x be object such that
A7: x in dom f \/ dom g;
    thus x in dom g implies h.x = g.x
    proof
      assume x in dom g;
      then x = b by TARSKI:def 1;
      hence thesis by A3,A4,FUNCOP_1:7;
    end;
    assume not x in dom g;
    then x in dom f by A7,XBOOLE_0:def 3;
    then x = a by TARSKI:def 1;
    hence h.x = f.x by A2,A5,FUNCOP_1:7;
  end;
  dom h = dom f \/ dom g by A1,ENUMSET1:1;
  hence thesis by A6,Def1;
end;
