theorem
  dom f = dom g & rng f = {y} & rng g = {y} implies f = g
proof
  assume that
A1: dom f = dom g and
A2: rng f = {y} and
A3: rng g = {y};
  x in dom f implies f.x = g.x
  proof
    assume
A4: x in dom f;
    then f.x in rng f by Def3;
    then
A5: f.x = y by A2,TARSKI:def 1;
    g.x in rng g by A1,A4,Def3;
    hence thesis by A3,A5,TARSKI:def 1;
  end;
  hence thesis by A1,Th2;
end;
