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

theorem Th8:
  for US being non void UniformSpaceStr holds
  US is axiom_U1 iff for S being Element of the entourages of US
  ex R being total reflexive Relation of the carrier of US st R = S
  proof
    let US be non void UniformSpaceStr;
    US is non void;
    then reconsider SFX = the entourages of US as non empty
      Subset-Family of [:the carrier of US,the carrier of US:];
    hereby
      assume
A2:   US is axiom_U1;
      hereby
        let S being Element of the entourages of US;
        consider R being Relation of the carrier of US such that
A3:     R = S and
A4:     R is_reflexive_in the carrier of US by A2,Th7;
A5:     field R = the carrier of US by A4,PARTIT_2:9;
        reconsider R as total reflexive Relation of the carrier of US
          by A5,A4,TAXONOM1:3,PARTFUN1:def 2,RELAT_2:def 9;
        take R;
        thus R = S by A3;
      end;
    end;
    assume
A6: for S being Element of the entourages of US holds
    ex R being total reflexive Relation of the carrier of US st R = S;
    now
      let S being Element of the entourages of US;
      consider R being total reflexive Relation of the carrier of US such that
A7:   R = S by A6;
      reconsider R as Relation of the carrier of US;
      take R;
      thus R = S by A7;
      now
        let x be object;
        assume
A8:     x in the carrier of US;
        field R = the carrier of US by ORDERS_1:12;
        hence [x,x] in R by A8,RELAT_2:def 9,RELAT_2:def 1;
      end;
      hence R is_reflexive_in the carrier of US by RELAT_2:def 1;
    end;
    hence US is axiom_U1 by Th7;
end;
