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
  for R being Tolerance of X holds
  RelStr2UniformSpaceStr(UniformSpaceStr2RelStr(uniformity_induced_by(R)))
    = uniformity_induced_by(R)
  proof
    let R be Tolerance of X;
    the carrier of
      RelStr2UniformSpaceStr(UniformSpaceStr2RelStr(uniformity_induced_by(R)))
    = the carrier of uniformity_induced_by(R) &
    the entourages of
      RelStr2UniformSpaceStr(UniformSpaceStr2RelStr(uniformity_induced_by(R)))
    = rho (meet(the entourages of uniformity_induced_by(R)));
    hence thesis by Th28;
  end;
