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 Th29:
  for SF being Subset-Family of [:X,X:],R being Relation of X st
  SF = rho(R) holds SF c= rho(meet SF)
  proof
    let SF be Subset-Family of [:X,X:],R be Relation of X;
    assume
A1: SF = rho(R);
    SF c= rho(meet(SF))
    proof
      let x be object;
      assume
A2:   x in SF;
      then consider S be Subset of [:X,X:] such that
A3:   x = S and
      R c= S by A1;
      meet(SF) c= S by A3,A2,SETFAM_1:def 1;
      hence thesis by A3;
    end;
    hence thesis;
  end;
