
theorem Th10:
  for G1, G2, G3, G4 being _Graph, F being PGraphMapping of G1, G2
  st (ex E1, E2 being set st G3 is reverseEdgeDirections of G1, E1 &
    G4 is reverseEdgeDirections of G2, E2)
  holds F is PGraphMapping of G3, G4
proof
  let G1, G2, G3, G4 being _Graph, F be PGraphMapping of G1, G2;
  given E1, E2 being set such that
    A1: G3 is reverseEdgeDirections of G1, E1 &
      G4 is reverseEdgeDirections of G2, E2;
  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 A1, GLIB_007:4;
  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 A3: e in dom g;
      then A4: (the_Source_of G1).e in dom f & (the_Target_of G1).e in dom f
        by Th5;
      e in the_Edges_of G1 by A2, A3;
      then A5: e DJoins (the_Source_of G1).e, (the_Target_of G1).e, G1
        by GLIB_000:def 14;
      per cases;
      suppose A6: E1 c= the_Edges_of G1;
        per cases;
        suppose e in E1;
          then e DJoins (the_Target_of G1).e, (the_Source_of G1).e, G3
            by A1, A5, A6, GLIB_007:7;
          hence (the_Source_of G3).e in dom f & (the_Target_of G3).e in dom f
            by A4, GLIB_000:def 14;
        end;
        suppose not e in E1;
          then e DJoins (the_Source_of G1).e, (the_Target_of G1).e, G3
            by A1, A5, A6, GLIB_007:8;
          hence (the_Source_of G3).e in dom f & (the_Target_of G3).e in dom f
            by A4, GLIB_000:def 14;
        end;
      end;
      suppose not E1 c= the_Edges_of G1;
        then G1 == G3 by A1, GLIB_007:def 1;
        hence (the_Source_of G3).e in dom f & (the_Target_of G3).e in dom f
          by A4, GLIB_000:def 34;
      end;
    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_007:9;
  end;
  then [f,g] is PGraphMapping of G3, G4 by Th8;
  hence thesis;
end;
