
theorem Th89:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v being Vertex of G1 st F is onto semi-Dcontinuous & v in dom F_V holds
    F_E.:(v.edgesIn()) = (F_V/.v).edgesIn() &
    F_E.:(v.edgesOut()) = (F_V/.v).edgesOut()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2, v be Vertex of G1;
  assume A1: F is onto semi-Dcontinuous & v in dom F_V;
  then A2: F_E.:(v.edgesIn()) c= (F_V/.v).edgesIn() &
    F_E.:(v.edgesOut()) c= (F_V/.v).edgesOut() by Th87;
  now
    let e9 be object;
    assume A3: e9 in (F_V/.v).edgesIn();
    then e9 in the_Edges_of G2;
    then e9 in rng F_E by A1, GLIB_010:def 12;
    then consider e being object such that
      A4: e in dom F_E & F_E.e = e9 by FUNCT_1:def 3;
    A5: (the_Target_of G2).e9 = F_V/.v by A3, GLIB_000:56;
    set w9 = (the_Source_of G2).e9;
    A6: F_E.e DJoins w9,F_V/.v,G2 by A3, A4, A5, GLIB_000:def 14;
    then F_E.e Joins w9,F_V/.v,G2 by GLIB_000:16;
    then w9 in the_Vertices_of G2 by GLIB_000:13;
    then w9 in rng F_V by A1, GLIB_010:def 12;
    then consider w being object such that
      A7: w in dom F_V & F_V.w = w9 by FUNCT_1:def 3;
    F_E.e DJoins F_V.w,F_V.v,G2 by A1, A6, A7, PARTFUN1:def 6;
    then A8: e DJoins w,v,G1 by A1, A4, A7, GLIB_010:def 17;
    e is set & w is set by TARSKI:1;
    then e in v.edgesIn() by A8, GLIB_000:57;
    hence e9 in F_E.:(v.edgesIn()) by A4, FUNCT_1:def 6;
  end;
  then (F_V/.v).edgesIn() c= F_E.:(v.edgesIn()) by TARSKI:def 3;
  hence F_E.:(v.edgesIn()) = (F_V/.v).edgesIn() by A2, XBOOLE_0:def 10;
  now
    let e9 be object;
    assume A9: e9 in (F_V/.v).edgesOut();
    then e9 in the_Edges_of G2;
    then e9 in rng F_E by A1, GLIB_010:def 12;
    then consider e being object such that
      A10: e in dom F_E & F_E.e = e9 by FUNCT_1:def 3;
    A11: (the_Source_of G2).e9 = F_V/.v by A9, GLIB_000:58;
    set w9 = (the_Target_of G2).e9;
    A12: F_E.e DJoins F_V/.v,w9,G2 by A9, A10, A11, GLIB_000:def 14;
    then F_E.e Joins F_V/.v,w9,G2 by GLIB_000:16;
    then w9 in the_Vertices_of G2 by GLIB_000:13;
    then w9 in rng F_V by A1, GLIB_010:def 12;
    then consider w being object such that
      A13: w in dom F_V & F_V.w = w9 by FUNCT_1:def 3;
    F_E.e DJoins F_V.v,F_V.w,G2 by A1, A12, A13, PARTFUN1:def 6;
    then A14: e DJoins v,w,G1 by A1, A10, A13, GLIB_010:def 17;
    e is set & w is set by TARSKI:1;
    then e in v.edgesOut() by A14, GLIB_000:59;
    hence e9 in F_E.:(v.edgesOut()) by A10, FUNCT_1:def 6;
  end;
  then (F_V/.v).edgesOut() c= F_E.:(v.edgesOut()) by TARSKI:def 3;
  hence F_E.:(v.edgesOut()) = (F_V/.v).edgesOut() by A2, XBOOLE_0:def 10;
end;
