
theorem Th25:
  for G1, G2 being _Graph, F being Dcontinuous PGraphMapping of G1, G2
  st F_E is one-to-one holds F is directed
proof
  let G1, G2 be _Graph;
  let F be Dcontinuous PGraphMapping of G1, G2;
  assume A1: F_E is one-to-one;
  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 A3: e DJoins v,w,G1;
    then e Joins v,w,G1 by GLIB_000:16;
    then per cases by A2, Th4, GLIB_000:16;
    suppose F_E.e DJoins F_V.v,F_V.w,G2;
      hence F_E.e DJoins F_V.v,F_V.w,G2;
    end;
    suppose A4: F_E.e DJoins F_V.w,F_V.v,G2;
      then consider e0 being object such that
        A5: e0 DJoins w,v,G1 & e0 in dom F_E & F_E.e0 = F_E.e by A2, Def18;
      e0 = e by A1, A2, A5, FUNCT_1:def 4;
      then v = w by A3, A5, GLIB_000:125;
      hence F_E.e DJoins F_V.v,F_V.w,G2 by A4;
    end;
  end;
  hence F is directed;
  thus thesis;
end;
