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 Th39:
  a is not Integer & [\a/]+1<b implies
   ex u be Integer st |.a-u.|<1 & |.a-u.|*|.b-u.|<|.a-b.|/2
   proof
     assume that
A1:  a is not Integer and
A2:  [\a/]+1 < b;
     assume
A3:  not ex u be Integer st |.a-u.|<1 & |.a-u.|*|.b-u.| < |.a-b.|/2;
     set u=[\a/],v=u+1;
A4:  |.a-u.|<1 by A1,Th21;
     |.a-v.|<1 by A1,Th21; then
b1:  |.a-v.|*|.b-v.| >= |.a-b.|/2 by A3;
     |.a-u.|*|.b-u.| >= |.a-b.|/2 by A3,A4;
     hence thesis by A1,A2,Th37,b1;
   end;
