
theorem
  for G3 being _Graph, G4 being G3-isomorphic _Graph, v1, v2 being object
  for G1 being addAdjVertexAll of G3,v1
  for G2 being addAdjVertexAll of G4,v2
  st v1 in the_Vertices_of G3 iff v2 in the_Vertices_of G4
  holds G2 is G1-isomorphic
proof
  let G3 be _Graph, G4 be G3-isomorphic _Graph, v1, v2 be object;
  let G1 be addAdjVertexAll of G3, v1;
  let G2 be addAdjVertexAll of G4, v2;
  assume v1 in the_Vertices_of G3 iff v2 in the_Vertices_of G4;
  then per cases;
  suppose A1: not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4;
    consider F0 being PGraphMapping of G3, G4 such that
      A2: F0 is isomorphism by Def23;
    A3: F0_V | the_Vertices_of G3 is one-to-one by A2;
    A4: dom(F0_V | the_Vertices_of G3) = the_Vertices_of G3 by A2, Def11;
    rng(F0_V | the_Vertices_of G3) = the_Vertices_of G4 by A2, Def12;
    then consider F being PGraphMapping of G1, G2 such that
      F_V = F0_V +* (v1 .--> v2) & F_E | dom F0_E = F0_E and
      F0 is total implies F is total and
      F0 is onto implies F is onto and
      F0 is one-to-one implies F is one-to-one and
      F0 is weak_SG-embedding implies F is weak_SG-embedding and
      A5: F0 is isomorphism implies F is isomorphism by A1, A3, A4, Th162;
    thus thesis by A2, A5;
  end;
  suppose v1 in the_Vertices_of G3 & v2 in the_Vertices_of G4;
    then G1 == G3 & G2 == G4 by GLIB_007:def 4;
    then G1 is reverseEdgeDirections of G3, {} &
      G2 is reverseEdgeDirections of G4, {} by GLIB_009:42;
    hence thesis by Th143;
  end;
end;
