reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th16:
  t > 0 implies ex i st t*i <= r <= t*(i+1)
  proof
    assume
A0: t > 0;
    defpred P[Integer] means t*$1 <= r;
    g: ex i1 being Integer st P[i1]
    proof
      take i1=[\r/t/];
      i1 <= r/t by INT_1:def 6;
      then i1*t <= r/t*t by A0,XREAL_1:64;
      then t*i1 <= r/(t/t) by XCMPLX_1:82;
      hence t*i1 <= r by A0,XCMPLX_1:51;
    end;
    set F=[/r/t\];
    f: for i1 being Integer st P[i1] holds i1 <= F
    proof
      let i1 be Integer;
      assume P[i1];
      then i1*t/t <= r/t by A0,XREAL_1:72;
      then i1*(t/t) <= r/t;
      then i1*1 <= r/t & r/t <= [/r/t\] by A0,XCMPLX_1:60,INT_1:def 7;
      hence i1 <= F by XXREAL_0:2;
    end;
    consider i such that
    i: P[i] & for i1 being Integer st P[i1] holds i1<=i from INT_1:sch 6(f,g);
    take i;
    thus t*i <= r by i;
    i+1 > i+0 by XREAL_1:6;
    hence r <= t*(i+1) by i;
  end;
