
theorem Th1:
  for G1, G2 being _Graph
  for f being PartFunc of the_Vertices_of G1, the_Vertices_of G2
  holds f is PVertexMapping of G1, G2 iff
    for v,w,e being object st v in dom f & w in dom f & e Joins v,w,G1
    ex e9 being object st e9 Joins f.v,f.w,G2
proof
  let G1, G2 be _Graph;
  let f be PartFunc of the_Vertices_of G1, the_Vertices_of G2;
  hereby
    assume A1: f is PVertexMapping of G1, G2;
    let v,w,e be object;
    assume A2: v in dom f & w in dom f & e Joins v,w,G1;
    then reconsider v0 = v, w0 = w as Vertex of G1;
    v0, w0 are_adjacent by A2, CHORD:def 3;
    then consider e9 being object such that
      A3: e9 Joins f/.v0, f/.w0, G2 by A1, A2, Def1, CHORD:def 3;
    take e9;
    f/.v0 = f.v & f/.w0 = f.w by A2, PARTFUN1:def 6;
    hence e9 Joins f.v,f.w,G2 by A3;
  end;
  assume A4: for v,w,e being object st v in dom f & w in dom f & e Joins v,w,G1
    ex e9 being object st e9 Joins f.v,f.w,G2;
  let v,w be Vertex of G1;
  assume A5: v in dom f & w in dom f;
  assume v,w are_adjacent;
  then consider e being object such that
    A6: e Joins v,w,G1 by CHORD:def 3;
  consider e9 being object such that
    A7: e9 Joins f.v,f.w,G2 by A4, A5, A6;
  f/.v = f.v & f/.w = f.w by A5, PARTFUN1:def 6;
  hence thesis by A7, CHORD:def 3;
end;
