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

theorem Th7:
  for US being non void UniformSpaceStr holds
  US is axiom_U1 iff for S being Element of the entourages of US holds
  ex R being Relation of the carrier of US st R = S &
  R is_reflexive_in the carrier of US
  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
A1:   US is axiom_U1;
      now
        let S be Element of the entourages of US;
        S is Element of SFX;
        then consider R be Relation of the carrier of US such that
A2:     R = S;
        take R;
        thus R = S by A2;
        now
          let x be object;
          assume x in the carrier of US; then
A3:       [x,x] in id the carrier of US by RELAT_1:def 10;
          id the carrier of US c= S by A1;
          hence [x,x] in R by A3,A2;
        end;
        hence R is_reflexive_in the carrier of US by RELAT_2:def 1;
      end;
      hence for S being Element of the entourages of US holds
        ex R being Relation of the carrier of US st (R = S &
        R is_reflexive_in the carrier of US);
    end;
    assume
A4: for S being Element of the entourages of US ex R being Relation of
      the carrier of US st (R = S & R is_reflexive_in the carrier of US);
    now
      let S be Element of the entourages of US;
      consider R be Relation of the carrier of US such that
A5:   R = S and
A6:   R is_reflexive_in the carrier of US by A4;
      for x be object st x in the carrier of US holds [x,x] in R
        by A6,RELAT_2:def 1;
      hence id the carrier of US c= S by A5,RELAT_1:47;
    end;
    hence US is axiom_U1;
  end;
