reserve G,G1,G2 for _Graph;
reserve e,x,y for set;
reserve v,v1,v2 for Vertex of G;
reserve W for Walk of G;

theorem
  G is Component of G iff G is connected
proof
  thus G is Component of G implies G is connected;
A1: G is Subgraph of G by GLIB_000:40;
A2: now
    given G2 being connected Subgraph of G such that
A3: G c< G2;
    now
      per cases by A1,A3,GLIB_000:98;
      suppose
        the_Vertices_of G c< the_Vertices_of G2;
        hence contradiction by XBOOLE_0:def 8;
      end;
      suppose
        the_Edges_of G c< the_Edges_of G2;
        hence contradiction by XBOOLE_0:def 8;
      end;
    end;
    hence contradiction;
  end;
  assume G is connected;
  hence thesis by A2,Def7,GLIB_000:40;
end;
