reserve G for _Graph;

theorem Th24:
  for G being non _trivial _Graph, v being Vertex of G
  for H being removeVertex of G, v st v is isolated
  holds VertexDomRel(H) = VertexDomRel(G)
proof
  let G be non _trivial _Graph, v be Vertex of G;
  let H be removeVertex of G, v;
  assume A1: v is isolated;
  set V1 = [: {v}, the_Vertices_of G :];
  set V2 = [: the_Vertices_of G, {v} :];
  (V1 \/ V2) /\ VertexDomRel(G) = {}
  proof
    assume (V1 \/ V2) /\ VertexDomRel(G) <> {};
    then consider z being object such that
      A2: z in (V1 \/ V2) /\ VertexDomRel(G) by XBOOLE_0:def 1;
    consider u,w being object such that
      A3: z = [u,w] by A2, RELAT_1:def 1;
    A4: [u,w] in V1 \/ V2 & [u,w] in VertexDomRel(G) by A2, A3, XBOOLE_0:def 4;
    then consider e being object such that
      A5: e DJoins u,w,G by Th1;
    per cases by A4, XBOOLE_0:def 3;
    suppose [u,w] in V1;
      then u in {v} by ZFMISC_1:87;
      then u = v by TARSKI:def 1;
      hence contradiction by A1, A5, GLIB_000:144;
    end;
    suppose [u,w] in V2;
      then w in {v} by ZFMISC_1:87;
      then w = v by TARSKI:def 1;
      hence contradiction by A1, A5, GLIB_000:144;
    end;
  end;
  then VertexDomRel(G) = VertexDomRel(G) \ (V1 \/ V2)
    by XBOOLE_1:83,XBOOLE_0:def 7;
  hence thesis by Th23;
end;
