
theorem Th146:
  for G3 being _Graph, G4 being G3-isomorphic _Graph, V1, V2 being set
  for G1 being addVertices of G3,V1, G2 being addVertices of G4,V2
  st card(V1 \ the_Vertices_of G3) = card(V2 \ the_Vertices_of G4)
  holds G2 is G1-isomorphic
proof
  let G3 be _Graph, G4 be G3-isomorphic _Graph, V1, V2 be set;
  let G1 be addVertices of G3,V1, G2 be addVertices of G4,V2;
  assume card(V1 \ the_Vertices_of G3) = card(V2 \ the_Vertices_of G4);
  then consider f being Function such that
    A1: f is one-to-one & dom f = V1 \ the_Vertices_of G3 &
      rng f = V2 \ the_Vertices_of G4 by CARD_1:5, WELLORD2:def 4;
  reconsider f as one-to-one Function by A1;
  consider F0 be PGraphMapping of G3, G4 such that
    A2: F0 is isomorphism by Def23;
  consider F being PGraphMapping of G1, G2 such that
    F = [F0_V +* f, F0_E] and
    F0 is weak_SG-embedding implies F is weak_SG-embedding and
    F0 is strong_SG-embedding implies F is strong_SG-embedding and
    A3: F0 is isomorphism implies F is isomorphism and
    F0 is Disomorphism implies F is Disomorphism by A1, Th145;
  thus thesis by A2, A3;
end;
