
theorem Th32:
  for G1 being _finite real-weighted WGraph, n being Nat, G2 being
  inducedSubgraph of G1, (PRIM:CompSeq(G1).n)`1 holds G2 is connected
proof
  let G1 be _finite real-weighted WGraph, n be Nat;
  set V = (PRIM:CompSeq(G1).n)`1;
  let G2 be inducedSubgraph of G1, V;
  reconsider V as non empty Subset of the_Vertices_of G1 by Th30;
  set E = (PRIM:CompSeq(G1).n)`2;
  reconsider E as Subset of G1.edgesBetween(V) by Th30;
  set G3 = the inducedSubgraph of G1,V,E;
A1: the_Vertices_of G3 = V & the_Vertices_of G2 = V by GLIB_000:def 37;
  the_Edges_of G3 = E & the_Edges_of G2 = G1.edgesBetween(V) by GLIB_000:def 37
;
  then reconsider G3 as Subgraph of G2 by A1,GLIB_000:44;
A2: G3 is spanning by A1;
  G3 is connected by Th31;
  hence thesis by A2,GLIB_002:23;
end;
