
theorem
  for F1, F2 being non empty Graph-yielding Function st dom F1 = dom F2 &
    for x1 being Element of dom F1, x2 being Element of dom F2 st x1 = x2
    holds F2.x2 is F1.x1-isomorphic
  holds F1, F2 are_isomorphic
proof
  let F1, F2 be non empty Graph-yielding Function;
  assume A1: dom F1 = dom F2;
  assume A2: for x1 being Element of dom F1, x2 being Element of dom F2
    st x1 = x2 holds F2.x2 is F1.x1-isomorphic;
  reconsider p = id dom F1 as one-to-one Function;
  take p;
  thus dom p = dom F1 & rng p = dom F2 by A1;
  let x be object;
  assume A3: x in dom F1;
  then reconsider G1 = F1.x as _Graph;
  reconsider G2 = F2.x as _Graph by A1, A3;
  take G1, G2;
  thus G1 = F1.x & G2 = F2.(p.x) by A3, FUNCT_1:18;
  thus G2 is G1-isomorphic by A1, A2, A3;
end;
