
theorem
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for X,Y being Subset of the_Vertices_of G1 st F is isomorphism
  holds card G1.edgesBetween(X,Y) = card G2.edgesBetween(F_V.:X,F_V.:Y)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let X,Y be Subset of the_Vertices_of G1;
  assume A1: F is isomorphism;
  then A2: card G1.edgesBetween(X,Y) c= card G2.edgesBetween(F_V.:X,F_V.:Y)
    by Th46;
  reconsider F0 = F as one-to-one PGraphMapping of G1, G2 by A1;
  F0" is isomorphism by A1, Th75;
  then A3: card G2.edgesBetween(F_V.:X,F_V.:Y)
    c= card G1.edgesBetween(F0"_V.:(F_V.:X),F0"_V.:(F_V.:Y)) by Th46;
  A4: dom F_V = the_Vertices_of G1 by A1, Def11;
  A5: F0"_V.:(F_V.:X) = (F_V)"(F_V.:X) by FUNCT_1:85
    .= X by A1, A4, FUNCT_1:94;
  F0"_V.:(F_V.:Y) = (F_V)"(F_V.:Y) by FUNCT_1:85
    .= Y by A1, A4, FUNCT_1:94;
  hence thesis by A2, A3, A5, XBOOLE_0:def 10;
end;
