
theorem
  for G1, G2 being non-multi _Graph, f being PVertexMapping of G1, G2
  st f is continuous one-to-one holds PVM2PGM(f) is semi-continuous
proof
  let G1, G2 be non-multi _Graph, f be PVertexMapping of G1, G2;
  assume A1: f is continuous one-to-one;
  set g = (PVM2PGM f)_E;
  now
    let e,v,w be object;
    assume A2: e in dom g & v in dom (PVM2PGM f)_V & w in dom (PVM2PGM f)_V;
    assume A3: g.e Joins (PVM2PGM f)_V.v,(PVM2PGM f)_V.w,G2;
    then g.e Joins f.v,f.w,G2;
    then consider e0 being object such that
      A4: e0 Joins v,w,G1 by A1, A2, Th2;
    e0 in G1.edgesBetween(dom f) by A2, A4, GLIB_000:32;
    then A5: e0 in dom g by Def10;
    then g.e0 Joins (PVM2PGM f)_V.v,(PVM2PGM f)_V.w,G2 by A2, A4, GLIB_010:4;
    then A6: g.e0 = g.e by A3, GLIB_000:def 20;
    PVM2PGM(f) is one-to-one by A1, Th34;
    hence e Joins v,w,G1 by A2, A4, A5, A6, FUNCT_1:def 4;
  end;
  hence thesis by GLIB_010:def 15;
end;
