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 Th28:
  for R be Relation of X holds meet rho(R) = R
  proof
    let R be Relation of X;
    thus meet rho(R) c= R
    proof
      let x be object;
      assume
A2:   x in meet rho(R);
      R in rho(R);
      hence thesis by A2,SETFAM_1:def 1;
    end;
    let x be object;
    assume
A3: x in R;
    now
      let Y be set;
      assume Y in rho(R);
      then ex S be Relation of X st Y = S & R c= S;
      hence x in Y by A3;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
