
theorem
  for G1, G2 being _Graph, x,y being object
  holds x .--> G1, y .--> G2 are_isomorphic iff G2 is G1-isomorphic
proof
  let G1, G2 be _Graph, x,y be object;
  hereby
    assume x .--> G1, y .--> G2 are_isomorphic;
    then consider p being one-to-one Function such that
      A1: dom p = dom(x .--> G1) & rng p = dom(y .--> G2) and
      A2: for z being object st z in dom(x .--> G1) ex G3, G4 being _Graph
        st G3 = (x .--> G1).z & G4 = (y .--> G2).(p.z) & G4 is G3-isomorphic;
    dom p = dom{[x,G1]} & rng p = dom{[y,G2]} by A1, FUNCT_4:82;
    then dom p = {x} & rng p = {y} by RELAT_1:9;
    then A3: p = x .--> y by FUNCT_4:112;
    dom(x .--> G1) = dom{[x,G1]} by FUNCT_4:82
      .= {x} by RELAT_1:9;
    then x in dom(x .--> G1) by TARSKI:def 1;
    then consider G3, G4 being _Graph such that
      A4: G3 = (x .--> G1).x & G4 = (y .--> G2).(p.x) & G4 is G3-isomorphic
      by A2;
    A5: G3 = G1 by A4, FUNCOP_1:72;
    G4 = (y .--> G2).y by A3, A4, FUNCOP_1:72
      .= G2 by FUNCOP_1:72;
    hence G2 is G1-isomorphic by A4, A5;
  end;
  assume A6: G2 is G1-isomorphic;
  take p = x .--> y;
  thus dom p = dom{[x,y]} by FUNCT_4:82
    .= {x} by RELAT_1:9
    .= dom{[x,G1]} by RELAT_1:9
    .= dom(x .--> G1) by FUNCT_4:82;
  p = {x} --> y by FUNCOP_1:def 9;
  hence rng p = {y} by FUNCOP_1:8
    .= dom{[y,G2]} by RELAT_1:9
    .= dom(y .--> G2) by FUNCT_4:82;
  let z be object;
  assume z in dom(x .--> G1);
  then A7: z = x by TARSKI:def 1;
  take G1, G2;
  thus G1 = (x .--> G1).z by A7, FUNCOP_1:72;
  thus G2 = (y .--> G2).y by FUNCOP_1:72
    .= (y .--> G2).(p.z) by A7, FUNCOP_1:72;
  thus thesis by A6;
end;
