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 Th17:
   n <= b & b <= n+1 implies |.n-b.|*|.n+1-b.| <=1/4
   proof
     assume that
A1:  n <= b and
A2:  b <= n+1;
     set x=b-n,y=n+1-b;
     x >= 0 by A1,XREAL_1:48; then
A4:  |. n - b .| = -(-x) by ABSVALUE:30 .= x;
A5:  x + y = 1;
     |.n-b.|*|.n+1-b.| = x*y by A4,A2, XREAL_1:48,ABSVALUE:def 1;
     hence thesis by A5,SERIES_3:18;
   end;
