
theorem Th38:
  for G1 being _Graph, G2 being spanning Subgraph of G1
  st (for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
    holds G1.reachableFrom(v1) = G2.reachableFrom(v2))
  holds G1.numComponents() = G2.numComponents()
proof
  let G1 be _Graph, G2 be spanning Subgraph of G1;
  assume A1: for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
    holds G1.reachableFrom(v1) = G2.reachableFrom(v2);
  thus G1.numComponents() = card G1.componentSet() by GLIB_002:def 9
    .= card G2.componentSet() by A1, Th37
    .= G2.numComponents() by GLIB_002:def 9;
end;
