
theorem Th88:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v being Vertex of G1 st F is onto semi-continuous & v in dom F_V
  holds F_E.:(v.edgesInOut()) = (F_V/.v).edgesInOut()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2, v be Vertex of G1;
  assume A1: F is onto semi-continuous & v in dom F_V;
  then A2: F_E.:(v.edgesInOut()) c= (F_V/.v).edgesInOut() by Th86;
  now
    let e9 be object;
    assume A3: e9 in (F_V/.v).edgesInOut();
    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;
    per cases by A3, GLIB_000:61;
    suppose A5: (the_Source_of G2).e9 = F_V/.v;
      set w9 = (the_Target_of G2).e9;
      A6: F_E.e Joins F_V/.v,w9,G2 by A3, A4, A5, GLIB_000:def 13;
      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 Joins F_V.v,F_V.w,G2 by A1, A6, A7, PARTFUN1:def 6;
      then e in v.edgesInOut() by A1, A4, A7, GLIB_010:def 15, GLIB_000:62;
      hence e9 in F_E.:(v.edgesInOut()) by A4, FUNCT_1:def 6;
    end;
    suppose A8: (the_Target_of G2).e9 = F_V/.v;
      set w9 = (the_Source_of G2).e9;
      A9: F_E.e Joins F_V/.v,w9,G2 by A3, A4, A8, GLIB_000:def 13;
      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
        A10: w in dom F_V & F_V.w = w9 by FUNCT_1:def 3;
      F_E.e Joins F_V.v,F_V.w,G2 by A1, A9, A10, PARTFUN1:def 6;
      then e in v.edgesInOut() by A1, A4, A10, GLIB_010:def 15, GLIB_000:62;
      hence e9 in F_E.:(v.edgesInOut()) by A4, FUNCT_1:def 6;
    end;
  end;
  then (F_V/.v).edgesInOut() c= F_E.:(v.edgesInOut()) by TARSKI:def 3;
  hence thesis by A2, XBOOLE_0:def 10;
end;
