reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th14:
  for A being Subset of R^1 st A = RAT holds Cl A = the carrier of R^1
proof
  let A be Subset of R^1;
  assume
A1: A = RAT;
  the carrier of R^1 c= Cl A
  proof
    let x be object;
    assume x in the carrier of R^1;
    then reconsider p = x as Element of RealSpace by METRIC_1:def 13,TOPMETR:17
;
    now
      let r be Real;
      reconsider pr = p + r as Element of RealSpace by METRIC_1:def 13
,XREAL_0:def 1;
      assume r > 0;
      then consider Q being Rational such that
A2:   p < Q and
A3:   Q < pr by RAT_1:7,XREAL_1:29;
      reconsider P = Q as Element of RealSpace by METRIC_1:def 13,XREAL_0:def 1
;
      P - p < pr - p by A3,XREAL_1:9;
      then dist (p, P) < r by A2,Th13;
      then
A4:   P in Ball (p, r) by METRIC_1:11;
      Q in A by A1,RAT_1:def 2;
      hence Ball (p, r) meets A by A4,XBOOLE_0:3;
    end;
    hence thesis by GOBOARD6:92,TOPMETR:def 6;
  end;
  hence thesis by XBOOLE_0:def 10;
end;
