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 Th27:
  n+1 < b & n < a & a < n + 1 implies
    |.a-n.|*|.b-n.|*|.a-n-1.|*|.b-n-1.| <= |.a - b.|^2/4
   proof
     assume
A2:  n+1 < b & n < a & a < n + 1; then
A3:  n+1-b < 0 by XREAL_1:49;
A4:  n - a < 0 by A2,XREAL_1:49;
A5:  (1+n)-n < b-n by A2,XREAL_1:14;
A6:  n+1-a > 0 by A2,XREAL_1:50;
A7:  (n-a)*(n+1-b) > 0 by A3,A4;
     (b-n)*(n+1-a) > 0 by A5,A6;
     hence thesis by A7,Th26;
   end;
