reserve G for _Graph;

theorem Th27:
  for v, w being Vertex of G, e being object, H being addEdge of G,v,e,w
  st not e in the_Edges_of G holds VertexDomRel(H) = VertexDomRel(G) \/ {[v,w]}
proof
  let v,w be Vertex of G, e be object, H be addEdge of G,v,e,w;
  assume A1: not e in the_Edges_of G;
  now
    let x,y be object;
    hereby
      assume [x,y] in VertexDomRel(H);
      then consider e0 being object such that
        A2: e0 DJoins x,y,H by Th1;
      per cases by A2, GLIB_006:71;
      suppose e0 DJoins x,y,G;
        then [x,y] in VertexDomRel(G) by Th1;
        hence [x,y] in VertexDomRel(G) \/ {[v,w]} by XBOOLE_0:def 3;
      end;
      suppose A3: not e0 in the_Edges_of G;
        A4: the_Edges_of H = the_Edges_of G \/ {e} by A1, GLIB_006:def 11;
        e0 in the_Edges_of H by A2, GLIB_000:def 14;
        then e0 in {e} by A3, A4, XBOOLE_0:def 3;
        then A5: e DJoins x,y,H by A2, TARSKI:def 1;
        e DJoins v,w,H by A1, GLIB_006:105;
        then x = v & y = w by A5, GLIB_000:125;
        then [x,y] in {[v,w]} by TARSKI:def 1;
        hence [x,y] in VertexDomRel(G) \/ {[v,w]} by XBOOLE_0:def 3;
      end;
    end;
    assume [x,y] in VertexDomRel(G) \/ {[v,w]};
    then per cases by XBOOLE_0:def 3;
    suppose A6: [x,y] in VertexDomRel(G);
      G is Subgraph of H by GLIB_006:57;
      hence [x,y] in VertexDomRel(H) by A6, Th15, TARSKI:def 3;
    end;
    suppose [x,y] in {[v,w]};
      then [x,y] = [v,w] by TARSKI:def 1;
      hence [x,y] in VertexDomRel(H) by A1, Th1, GLIB_006:105;
    end;
  end;
  hence thesis by RELAT_1:def 2;
end;
