reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;
reserve v for object;

theorem Th137:
  for G2 for v1 being Vertex of G2, e,v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2
  st not v2 in the_Vertices_of G2 & not e in the_Edges_of G2
  for e1, w being object st (w <> v1 or e1 <> e)
  holds not e1 Joins w,v2,G1
proof
  let G2;
  let v1 be Vertex of G2, e be object, v2 be object;
  let G1 be addAdjVertex of G2,v1,e,v2;
  assume A1: not v2 in the_Vertices_of G2 & not e in the_Edges_of G2;
  let e1, w be object;
  assume w <> v1 or e1 <> e;
  then per cases;
  suppose A2: w <> v1;
    assume A3: e1 Joins w,v2,G1;
    then per cases by Th76;
    suppose e1 Joins w,v2,G2;
      hence contradiction by A1, GLIB_000:13;
    end;
    suppose A4: not e1 in the_Edges_of G2;
      A5: e1 in the_Edges_of G1 by A3, GLIB_000:def 13;
      the_Edges_of G1 = the_Edges_of G2 \/ {e} by A1, Def12;
      then e1 in {e} by A4, A5, XBOOLE_0:def 3;
      then e1 = e by TARSKI:def 1;
      then e1 Joins v1,v2,G1 by A1, Th135;
      then per cases by A3, GLIB_000:15;
      suppose v1=w & v2=v2;
        hence contradiction by A2;
      end;
      suppose v1=v2 & v2=w;
        hence contradiction by A1;
      end;
    end;
  end;
  suppose A6: e1 <> e;
    assume A7: e1 Joins w,v2,G1;
    then per cases by Th76;
    suppose e1 Joins w,v2,G2;
      hence contradiction by A1, GLIB_000:13;
    end;
    suppose A8: not e1 in the_Edges_of G2;
      A9: e1 in the_Edges_of G1 by A7, GLIB_000:def 13;
      the_Edges_of G1 = the_Edges_of G2 \/ {e} by A1, Def12;
      then e1 in {e} by A8, A9, XBOOLE_0:def 3;
      hence contradiction by A6, TARSKI:def 1;
    end;
  end;
end;
