reserve G for _Graph;

theorem
  for V being non empty Subset of the_Vertices_of G
  for H being inducedSubgraph of G, V
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G) /\ [: V, V :]
proof
  let V be non empty Subset of the_Vertices_of G;
  let H be inducedSubgraph of G, V;
  A1: VertexDomRel(H) = VertexDomRel(G) /\ [: V, V :] by Th21;
  then (VertexDomRel(H))~ = (VertexDomRel(G))~ /\ [: V, V :]~ by RELAT_1:22
    .= (VertexDomRel(G))~ /\ [: V, V :] by SYSREL:5;
  hence VertexAdjSymRel(H) = VertexAdjSymRel(G) /\ [: V, V :]
    by A1, XBOOLE_1:23;
end;
