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 Th22:
   |.n-a.|*|.n+1-a.| <=1/4 & |.n-b.|*|.n+1-b.| <=1/4 implies
   |.n-a.|*|.n-b.| <= 1/4 or |.n+1-a.|*|.n+1-b.| <=1/4
   proof
     assume
A1:  |.n-a.|*|.n+1-a.| <=1/4 & |.n-b.|*|.n+1-b.| <=1/4;
     set r1=|.n-a.|,r2=|.n-b.|,s1=|.n+1-a.|,s2=|.n+1-b.|;
A2:  (r1*s1)*(r2*s2) <= (1/4)*(1/4) by A1,XREAL_1:66;
     set r3=r1*r2, r4=s1*s2, r5=(1/4)*(1/4);
A3:  sqrt r5=(sqrt(1/4))^2 by SQUARE_1:29
     .= 1/4 by SQUARE_1:def 2;
     r3*r4 <= r5 by A2;
     hence thesis by A3,Th18;
   end;
