reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;
reserve c0,c1,c2,u,a0,b0 for Real;
reserve a,b for Real;
reserve n for Integer;

theorem Th28:
  a is not Integer & [\a/]<=b & b<=[\a/]+1 implies
    ex u be Integer st |.a-u.|<1 & |.a-u.|*|.b-u.|<= 1/4
   proof
     assume that
A1:  a is not Integer and
A2:  [\a/]<=b & b<=[\a/]+1;
     set n = [\a/];
A3:  n<a by A1,INT_1:26;
     a-1<n by INT_1:def 6; then
     a<n+1 by XREAL_1:19; then
A6:  |.n-a.|*|.n+1-a.| <=1/4 by Th17,A3;
     |.n-b.|*|.n+1-b.| <=1/4 by Th17,A2; then
     per cases by Th22,A6;
       suppose
A8:      |.n-a.|*|.n-b.| <= 1/4;
         set u = n;
A10:     |.a-u.|<1 by A1,Th21;
         |.u-a.|*|.u-b.| = |.a-u.|*|.u-b.| by COMPLEX1:60
         .= |.a-u.|*|.b-u.| by COMPLEX1:60;
         hence thesis by A10,A8;
       end;
       suppose
A12:      |.n+1-a.|*|.n+1-b.| <= 1/4;
         set u = n+1;
A14:     |.a-u.|<1 by A1,Th21;
         |.u-a.|*|.u-b.| = |.a-u.|*|.u-b.| by COMPLEX1:60
         .= |.a-u.|*|.b-u.| by COMPLEX1:60;
         hence thesis by A14,A12;
       end;
     end;
