
theorem Th16:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st for v,w being Vertex of G1 st v in dom F_V & w in dom F_V &
    v,w are_adjacent ex e being object st e in dom F_E & e Joins v,w,G1
  holds F_V is PVertexMapping of G1, G2
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: for v,w being Vertex of G1 st v in dom F_V & w in dom F_V &
    v,w are_adjacent ex e being object st e in dom F_E & e Joins v,w,G1;
  let v,w be Vertex of G1;
  assume A2: v in dom F_V & w in dom F_V & v,w are_adjacent;
  then consider e being object such that
    A3: e in dom F_E & e Joins v,w,G1 by A1;
  A4: F_V/.v = F_V.v & F_V/.w = F_V.w by A2, PARTFUN1:def 6;
  F_E.e Joins F_V.v,F_V.w,G2 by A2, A3, GLIB_010:4;
  hence F_V/.v, F_V/.w are_adjacent by A4, CHORD:def 3;
end;
