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;

theorem Th16:
   for rh0 be Element of REAL,p,q be Integer st p,q are_coprime holds
   ex x,y be Element of INT st |. p*x - q*y +rh0 .| <= 1/2
   proof
     let rh0 be Element of REAL;
     let p,q be Integer;
     assume
A1:  p,q are_coprime;
     consider b be Element of INT such that
A2:  |.rh0 - b .| <= 1/2 by GAUSSINT:48;
     consider s,t be Integer such that
A3:  s*p + t*q = 1 by A1,Th15;
A4:  b*(s*p + t*q) = b by A3;
     set x = (-b*s),y = b*t;
  X: x in INT & y in INT by INT_1:def 2;
     p*x -q*y + rh0 = rh0 - b by A4;
     hence thesis by A2,X;
   end;
