
theorem Th141:
  for G1, G2 being _Graph, E1, E2 being set
  for G3 being reverseEdgeDirections of G1, E1
  for G4 being reverseEdgeDirections of G2, E2
  for F0 being PGraphMapping of G1, G2
  ex F being PGraphMapping of G3, G4 st F = F0 &
    (F0 is non empty  implies F is non empty) &
    (F0 is total implies F is total) &
    (F0 is onto implies F is onto) &
    (F0 is one-to-one implies F is one-to-one) &
    (F0 is semi-continuous implies F is semi-continuous) &
    (F0 is continuous implies F is continuous)
proof
  let G1, G2 be _Graph, E1, E2 be set;
  let G3 be reverseEdgeDirections of G1, E1;
  let G4 be reverseEdgeDirections of G2, E2;
  let F0 be PGraphMapping of G1, G2;
  reconsider F = F0 as PGraphMapping of G3, G4 by Th10;
  take F;
  thus F = F0;
  thus F0 is non empty implies F is non empty;
  thus F0 is total implies F is total by GLIB_007:4;
  thus F0 is onto implies F is onto by GLIB_007:4;
  thus F0 is one-to-one implies F is one-to-one;
  thus F0 is semi-continuous implies F is semi-continuous
  proof
    assume A1: F0 is semi-continuous;
    now
      let e,v,w be object;
      assume A2: e in dom F_E & v in dom F_V & w in dom F_V;
      assume F_E.e Joins F_V.v, F_V.w, G4;
      then e Joins v,w,G1 by A1, A2, GLIB_007:9;
      hence e Joins v,w,G3 by GLIB_007:9;
    end;
    hence thesis;
  end;
  thus F0 is continuous implies F is continuous
  proof
    assume A3: F0 is continuous;
    now
      let e9,v,w be object;
      assume v in dom F_V & w in dom F_V & e9 Joins F_V.v, F_V.w, G4;
      then consider e being object such that
        A4: e Joins v,w,G1 & e in dom F0_E & F0_E.e = e9 by A3, GLIB_007:9;
      take e;
      thus e Joins v,w,G3 by A4, GLIB_007:9;
      thus e in dom F_E & F_E.e = e9 by A4;
    end;
    hence thesis;
  end;
end;
