
theorem Th39:
  for G1, G2 being non-multi _Graph, f being PVertexMapping of G1, G2
  st f is continuous holds PVM2PGM(f) is continuous
proof
  let G1, G2 be non-multi _Graph, f be PVertexMapping of G1, G2;
  assume A1: f is continuous;
  now
    let e9,v,w be object;
    assume A2: v in dom (PVM2PGM f)_V & w in dom (PVM2PGM f)_V;
    assume A3: e9 Joins (PVM2PGM f)_V.v,(PVM2PGM f)_V.w,G2;
    then consider e being object such that
      A4: e Joins v,w,G1 by A1, A2, Th2;
    take e;
    thus e Joins v,w,G1 by A4;
    e in G1.edgesBetween(dom (PVM2PGM f)_V) by A2, A4, GLIB_000:32;
    hence e in dom (PVM2PGM f)_E by Def10;
    then (PVM2PGM f)_E.e Joins (PVM2PGM f)_V.v,(PVM2PGM f)_V.w,G2
      by A2, A4, GLIB_010:4;
    hence (PVM2PGM f)_E.e = e9 by A3, GLIB_000:def 20;
  end;
  hence thesis by GLIB_010:def 16;
end;
