
theorem
  for G1 being _Graph, G2 being non-multi _Graph
  for F1, F2 being PGraphMapping of G1, G2
  st F1_V = F2_V & dom F1_E = dom F2_E holds F1 = F2
proof
  let G1 be _Graph, G2 be non-multi _Graph;
  let F1, F2 be PGraphMapping of G1, G2;
  assume that
    A1: F1_V = F2_V and
    A2: dom F1_E = dom F2_E;
  for e being object st e in dom F1_E holds F1_E.e = F2_E.e
  proof
    let e be object;
    set v = (the_Source_of G1).e, w = (the_Target_of G1).e;
    assume A3: e in dom F1_E;
    then A4: v in dom F1_V & w in dom F1_V by Th5;
    A5: e Joins v,w,G1 by A3, GLIB_000:def 13;
    then A6: F1_E.e Joins F1_V.v,F1_V.w,G2 by A3, A4, Th4;
    A7: e in dom F2_E by A2, A3;
    then v in dom F2_V & w in dom F2_V by Th5;
    then F2_E.e Joins F1_V.v,F2_V.w,G2 by A1, A5, A7, Th4;
    hence F1_E.e = F2_E.e by A1, A6, GLIB_000:def 20;
  end;
  then F1_E = F2_E by A2, FUNCT_1:2;
  hence thesis by A1, XTUPLE_0:2;
end;
