reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;
reserve US for UniformSpace;
reserve R for Relation of X;

theorem Th30:
  for SF being upper Subset-Family of [:X,X:] st meet SF in SF holds
  rho(meet SF) c= SF
  proof
    let SF be upper Subset-Family of [:X,X:];
    assume
A1: meet SF in SF;
    thus rho(meet(SF)) c= SF
    proof
      let x be object;
      assume x in rho(meet(SF));
      then consider S be Subset of [:X,X:] such that
A2:   x = S and
A3:   meet(SF) c= S;
      SF is upper;
      hence thesis by A2,A3,A1;
    end;
    thus thesis;
  end;
