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

theorem
  for G2 for v1,v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2
  st G2 is non complete & v1, v2 are_adjacent holds G1 is non complete
proof
  let G2;
  let v1, v2 be Vertex of G2, e be object;
  let G1 be addEdge of G2,v1,e,v2;
  assume that
    A1: G2 is non complete and
    A2: v1,v2 are_adjacent;
  per cases;
  suppose A3: not e in the_Edges_of G2;
    ex u1,u2 being Vertex of G1 st u1 <> u2 & not u1,u2 are_adjacent
    proof
      consider u3,u4 being Vertex of G2 such that
        A4: u3 <> u4 and
        A5: not u3,u4 are_adjacent by A1, CHORD:def 6;
      reconsider u1=u3, u2=u4 as Vertex of G1 by Th72;
      take u1, u2;
      thus u1 <> u2 by A4;
      not ex e1 being object st e1 Joins u1,u2,G1
      proof
        given e1 being object such that
          A6: e1 Joins u1,u2,G1;
        per cases by A6, Th76;
        suppose e1 Joins u1,u2,G2;
          hence contradiction by A5, CHORD:def 3;
        end;
        suppose A7: not e1 in the_Edges_of G2;
          A8: e1 in the_Edges_of G1 by A6, GLIB_000:def 13;
          the_Edges_of G1 = the_Edges_of G2 \/ {e} by A3, Def11;
          then e1 in {e} by A7, A8, XBOOLE_0:def 3;
          then e1 = e by TARSKI:def 1;
          then e1 DJoins v1,v2,G1 by A7, Th109;
          then e1 Joins v1,v2,G1 by GLIB_000:16;
          then per cases by A6, GLIB_000:15;
          suppose v1 = u1 & v2 = u2;
            hence contradiction by A2, A5;
          end;
          suppose v1 = u2 & v2 = u1;
            hence contradiction by A2, A5;
          end;
        end;
      end;
      hence thesis by CHORD:def 3;
    end;
    hence thesis by CHORD:def 6;
  end;
  suppose e in the_Edges_of G2;
    then G1 == G2 by Def11;
    hence thesis by A1, CHORD:62;
  end;
end;
