
theorem Th16:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  holds F is semi-Dcontinuous iff
    for e,v,w being object st e in dom F_E & v in dom F_V & w in dom F_V
    holds e DJoins v,w,G1 iff F_E.e DJoins F_V.v, F_V.w, G2
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  thus F is semi-Dcontinuous implies
    for e,v,w being object st e in dom F_E & v in dom F_V & w in dom F_V
      holds e DJoins v,w,G1 iff F_E.e DJoins F_V.v, F_V.w, G2
  proof
    assume A1: F is semi-Dcontinuous;
    let e,v,w be object;
    assume A2: e in dom F_E & v in dom F_V & w in dom F_V;
    thus e DJoins v,w,G1 implies F_E.e DJoins F_V.v, F_V.w, G2
    proof
      assume A3: e DJoins v,w,G1;
      then e Joins v,w,G1 by GLIB_000:16;
      then A4: F_E.e Joins F_V.v, F_V.w, G2 by A2, Th4;
      assume A5: not F_E.e DJoins F_V.v, F_V.w, G2;
      then A6: F_E.e DJoins F_V.w, F_V.v, G2 by A4, GLIB_000:16;
      then e DJoins w,v,G1 by A1, A2;
      then (the_Source_of G1).e = w & (the_Target_of G1).e = v
        by GLIB_000:def 14;
      then v = w by A3, GLIB_000:def 14;
      hence contradiction by A5, A6;
    end;
    thus thesis by A1, A2;
  end;
  thus thesis;
end;
