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 Th23:
   |.a-n.|*|.b-n.|*|.a-n-1.|*|.b-n-1.| <= |.a-b.|^2/4 implies
   |.a-n.|*|.b-n.|<=|.a-b.|/2 or |.a-n-1.|*|.b-n-1.|<=|.a-b.|/2
   proof
     assume
A1:  |.a-n.|*|.b-n.|*|.a-n-1.|*|.b-n-1.|<=|.a-b.|^2/4;
     set r1=|.a-n.|,r2=|.b-n.|,s1=|.a-n-1.|,s2=|.b-n-1.|;
     set t=a-b;
     set r0=|.a-b.|/2;
     set r3=r1*r2, r4=s1*s2;
A3:  sqrt(|.a-b.|^2/4)=sqrt((|.a-b.|/2)*(|.a-b.|/2))
     .=(sqrt(r0))^2 by SQUARE_1:29
     .=r0 by SQUARE_1:def 2;
     r3*r4 <= |.a-b.|^2/4 by A1;
     hence thesis by A3, Th18;
   end;
