reserve G for _Graph;

theorem Th47:
  for H being removeLoops of G
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G) \ id the_Vertices_of G
proof
  let H be removeLoops of G;
  A1: VertexDomRel(H) = VertexDomRel(G) \ id the_Vertices_of G by Th17;
  then (VertexDomRel(H))~
     = (VertexDomRel(G))~ \ (id the_Vertices_of G qua Relation)~ by RELAT_1:24
    .= (VertexDomRel(G))~ \ id the_Vertices_of G;
  hence thesis by A1, XBOOLE_1:42;
end;
