reserve X for set,
        A for Subset of X,
        R,S for Relation of X;

theorem
  for US being non void UniformSpaceStr st
  for S being Element of the entourages of US holds
  ex R being Tolerance of the carrier of US st S = R holds US is axiom_U1 &
  US is axiom_U2
  proof
    let US be non void UniformSpaceStr;
    assume
A2: for S being Element of the entourages of US holds
    ex R being Tolerance of the carrier of US st S = R;
    for S being Element of the entourages of US holds
    ex R being total reflexive Relation of the carrier of US st R = S
    proof
      let S be Element of the entourages of US;
      ex R being Tolerance of the carrier of US st R = S by A2;
      hence thesis;
    end;
    hence US is axiom_U1 by Th8;
    for S being Element of the entourages of US holds
    ex R being Relation of the carrier of US st S = R & R is symmetric
    proof
      let S be Element of the entourages of US;
      ex R being Tolerance of the carrier of US st S = R by A2;
      hence thesis;
    end;
    hence US is axiom_U2 by Th10;
  end;
