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;
