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 Th25:
   b<n & n<a & a<n+1 implies |.a-n.|*|.b-n.|*|.a-n-1.|*|.b-n-1.|<=|.a-b.|^2/4
   proof
     assume that
A1:  n>b and
A2:  n<a & a < n + 1;
A3:  n-b > 0 by A1,XREAL_1:50;
A4:  n+1 -a > 0 by A2, XREAL_1:50;
A5:  a-n > 0 by A2, XREAL_1:50;
A6:  (n-b) + 1 > 0 by A3;
A7:  (n-b)*(n+1-a) > 0 by A3,A4;
     (a-n)*(n+1-b) > 0 by A5,A6;
     hence thesis by A7,Th24;
   end;
