
theorem
  for G3 being _Graph, G4 being G3-Disomorphic _Graph, v1, v2 being object
  for G1 being addVertex of G3, v1, G2 being addVertex of G4, v2
  st v1 in the_Vertices_of G3 iff v2 in the_Vertices_of G4
  holds G2 is G1-Disomorphic
proof
  let G3 be _Graph, G4 be G3-Disomorphic _Graph, v1, v2 be object;
  let G1 be addVertex of G3,v1, G2 be addVertex of G4,v2;
  assume v1 in the_Vertices_of G3 iff v2 in the_Vertices_of G4;
  then per cases;
  suppose v1 in the_Vertices_of G3 & v2 in the_Vertices_of G4;
    then {v1} \ the_Vertices_of G3 = {} & {v2} \ the_Vertices_of G4 = {}
      by ZFMISC_1:60;
    then card({v1} \ the_Vertices_of G3) = card({v2} \ the_Vertices_of G4);
    hence thesis by Th147;
  end;
  suppose not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4;
    then {v1} \ the_Vertices_of G3 = {v1} & {v2} \ the_Vertices_of G4 = {v2}
      by ZFMISC_1:59;
    then card({v1} \ the_Vertices_of G3) = 1 &
      card({v2} \ the_Vertices_of G4) = 1 by CARD_1:30;
    hence thesis by Th147;
  end;
end;
