reserve G for _Graph;

theorem Th21:
  for V being non empty Subset of the_Vertices_of G
  for H being inducedSubgraph of G, V
  holds VertexDomRel(H) = VertexDomRel(G) /\ [: V, V :]
proof
  let V be non empty Subset of the_Vertices_of G;
  let H be inducedSubgraph of G, V;
  A1: the_Vertices_of H = V & the_Edges_of H = G.edgesBetween(V)
    by GLIB_000:def 37;
  now
    let v,w be object;
    hereby
      assume A2: [v,w] in VertexDomRel(H);
      then consider e being object such that
        A3: e DJoins v,w,H by Th1;
      e Joins v,w,H by A3, GLIB_000:16;
      then v in V & w in V by A1, GLIB_000:13;
      then A4: [v,w] in [: V, V :] by ZFMISC_1:87;
      [v,w] in VertexDomRel(G) by A2, Th15, TARSKI:def 3;
      hence [v,w] in VertexDomRel(G) /\ [: V, V :] by A4, XBOOLE_0:def 4;
    end;
    assume [v,w] in VertexDomRel(G) /\ [: V, V :];
    then A5: [v,w] in VertexDomRel(G) & [v,w] in [: V, V :] by XBOOLE_0:def 4;
    then consider e being object such that
      A6: e DJoins v,w,G by Th1;
    A7: e in the_Edges_of G & (the_Source_of G).e = v &
      (the_Target_of G).e = w by A6, GLIB_000:def 14;
    v in V & w in V by A5, ZFMISC_1:87;
    then A8: e in the_Edges_of H by A1, A7, GLIB_000:31;
    e is set & v is set & w is set by TARSKI:1;
    then e DJoins v,w,H by A6, A8, GLIB_000:73;
    hence [v,w] in VertexDomRel(H) by Th1;
  end;
  hence thesis by RELAT_1:def 2;
end;
