reserve G for _Graph;

theorem Th52:
  for V being set, H being removeVertices of G, V st V c< the_Vertices_of G
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G)
    \ ([: V, the_Vertices_of G :] \/ [: the_Vertices_of G, V :])
proof
  let V be set, H be removeVertices of G, V;
  assume A1: V c< the_Vertices_of G;
  set V1 = [: V, the_Vertices_of G :], V2 = [: the_Vertices_of G, V :];
  A2: VertexDomRel(H) = VertexDomRel(G) \ (V1 \/ V2) by A1, Th22;
  then (VertexDomRel(H))~ = (VertexDomRel(G))~ \ ((V1 \/ V2)~) by RELAT_1:24
    .= (VertexDomRel(G))~ \ (V1~ \/ V2~) by RELAT_1:23
    .= (VertexDomRel(G))~ \ (V1~ \/ V1) by SYSREL:5
    .= (VertexDomRel(G))~ \ (V2 \/ V1) by SYSREL:5;
  hence thesis by A2, XBOOLE_1:42;
end;
