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 Th31:
  for US being upper UniformSpaceStr st
  ex R being Relation of the carrier of US st
  the entourages of US = rho(R) &
  meet the entourages of US in the entourages of US holds
  the entourages of US = rho(meet the entourages of US)
  proof
    let US be upper UniformSpaceStr;
    given R being Relation of the carrier of US such that
A2: the entourages of US = rho(R) and
A3: meet the entourages of US in the entourages of US;
    the entourages of US is upper by UNIFORM2:def 7;
    hence thesis by A2,A3,Th29,Th30;
  end;
