
theorem Th9:
  for G1, G2, G3, G4 being _Graph, F being PGraphMapping of G1, G2
  st G1 == G3 & G2 == G4 holds F is PGraphMapping of G3, G4
proof
  let G1, G2, G3, G4 being _Graph, F be PGraphMapping of G1, G2;
  assume A1: G1 == G3 & G2 == G4;
  then A2: the_Vertices_of G1 = the_Vertices_of G3 &
    the_Edges_of G1 = the_Edges_of G3 &
    the_Vertices_of G2 = the_Vertices_of G4 &
    the_Edges_of G2 = the_Edges_of G4 by GLIB_000:def 34;
  then reconsider
    f = F_V as PartFunc of the_Vertices_of G3, the_Vertices_of G4;
  reconsider
    g = F_E as PartFunc of the_Edges_of G3, the_Edges_of G4 by A2;
  now
    hereby
      let e be object;
      assume e in dom g;
      then (the_Source_of G1).e in dom f & (the_Target_of G1).e in dom f
        by Th5;
      hence (the_Source_of G3).e in dom f & (the_Target_of G3).e in dom f
        by A1, GLIB_000:def 34;
    end;
    let e,v,w be object;
    assume e in dom g & v in dom f & w in dom f;
    then e Joins v,w,G1 implies g.e Joins f.v,f.w,G2 by Th4;
    hence e Joins v,w,G3 implies g.e Joins f.v,f.w,G4 by A1, GLIB_000:88;
  end;
  then [f,g] is PGraphMapping of G3, G4 by Th8;
  hence thesis;
end;
