
theorem Th21:
  for z,e be Real st 0 < e holds
  ex q be Element of RAT st q <> 0 & |. z - q .| < e
  proof
    let z be Real,e be Real;
    assume
    A1: 0 < e; then
    0 + z < z + e by XREAL_1:8; then
    consider p1, p2 being Rational such that
    A2: z < p1 & p1 < p2 & p2 < z + e by BORSUK_5:26;
    per cases;
    suppose
      A3: 0 <= z;
      p1 < z + e by A2,XXREAL_0:2; then
      p1 - z < z + e - z by XREAL_1:14; then
      A4: |. p1 - z .| < e by ABSVALUE:def 1,A2,XREAL_1:48;
      reconsider p1 as Element of RAT by RAT_1:def 2;
      take p1;
      thus p1 <> 0 by A2,A3;
      thus |. z - p1 .| < e by A4,COMPLEX1:60;
    end;
    suppose
      A5: z < 0;
      z - e < z - 0 by A1,XREAL_1:15; then
      consider p1,p2 being Rational such that
      A6: z - e < p1 & p1 < p2 & p2 < z by BORSUK_5:26;
      z - e - z < p1 - z by A6,XREAL_1:14; then
      A7: 0 - (p1 - z) < 0 - (-e) by XREAL_1:15;
      A8: p1 < z by A6,XXREAL_0:2;
      reconsider p1 as Element of RAT by RAT_1:def 2;
      take p1;
      thus p1 <> 0 by A5,A6;
      thus |. z - p1 .| < e by A7,A8,ABSVALUE:def 1,XREAL_1:48;
    end;
  end;
