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 Th41:
   ex u be Integer st |.a - u.| < 1 &
   (|.a - u .|*|. b - u .| <= 1/4 or |.a - u.|*|.b - u.| < |.a - b .|/2)
   proof
     per cases;
       suppose A: a in INT;
         |.a - a.|*|.b - a.| = 0;
         hence thesis by A;
       end;
       suppose not a in INT; then
A5:       a is not integer;
         per cases;
         suppose [\a/] <= b & b <= [\a/]+1; then
           ex u be Integer st |.a-u.|<1 & |.a-u.|*|.b-u.|<= 1/4 by Th28,A5;
           hence thesis;
         end;
         suppose b < [\a/]; then
           ex u be Integer st |.a-u.|<1 & |.a-u.|*|.b-u.| < |.a-b.|/2
             by Th33,A5;
           hence thesis;
         end;
         suppose [\a/]+1 < b; then
           ex u be Integer st |.a-u.|<1 & |.a-u.|*|.b-u.| < |.a-b.|/2
             by Th39,A5;
           hence thesis;
         end;
        end;
     end;
