
theorem Th34:
  for G1, G2 being non-multi _Graph, f being PVertexMapping of G1, G2
  st f is one-to-one holds PVM2PGM(f) is one-to-one
proof
  let G1, G2 be non-multi _Graph, f be PVertexMapping of G1, G2;
  assume A1: f is one-to-one;
  then A2: (PVM2PGM f)_V is one-to-one;
  set g = (PVM2PGM f)_E;
  for x1,x2 being object st x1 in dom g & x2 in dom g & g.x1 = g.x2
    holds x1 = x2
  proof
    let x1,x2 be object;
    set v1 = (the_Source_of G1).x1, w1 = (the_Target_of G1).x1,
      v2 = (the_Source_of G1).x2, w2 = (the_Target_of G1).x2;
    assume A3: x1 in dom g & x2 in dom g & g.x1 = g.x2;
    then x1 in G1.edgesBetween(dom f) by Def10;
    then A4: x1 in the_Edges_of G1 & v1 in dom f & w1 in dom f by GLIB_000:31;
    then A5: x1 Joins v1,w1,G1 by GLIB_000:def 13;
    then g.x1 Joins (PVM2PGM f)_V.v1,(PVM2PGM f)_V.w1,G2 by A3, A4, GLIB_010:4;
    then A6: g.x1 Joins f.v1,f.w1,G2;
    x2 in G1.edgesBetween(dom f) by A3, Def10;
    then A7: x2 in the_Edges_of G1 & v2 in dom f & w2 in dom f by GLIB_000:31;
    then A8: x2 Joins v2,w2,G1 by GLIB_000:def 13;
    then g.x2 Joins (PVM2PGM f)_V.v2,(PVM2PGM f)_V.w2,G2 by A3, A7, GLIB_010:4;
    then per cases by A3, A6, GLIB_000:15;
    suppose f.v1 = f.v2 & f.w1 = f.w2;
      then v1 = v2 & w1 = w2 by A1, A4, A7, FUNCT_1:def 4;
      hence x1 = x2 by A5, A8, GLIB_000:def 20;
    end;
    suppose f.v1 = f.w2 & f.w1 = f.v2;
      then v1 = w2 & w1 = v2 by A1, A4, A7, FUNCT_1:def 4;
      then x2 Joins v1,w1,G1 by A8, GLIB_000:14;
      hence x1 = x2 by A5, GLIB_000:def 20;
    end;
  end;
  then g is one-to-one by FUNCT_1:def 4;
  hence thesis by A2, GLIB_010:def 13;
end;
