reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem Th14:
  PI/2+2*PI*i < r & r < 3/2*PI+2*PI*i implies cos r < 0
proof
  assume that
A1: PI/2+T(i) < r and
A2: r < 3/2*PI+T(i);
  PI/2+T(i)+PI/2 < r+PI/2 by A1,XREAL_1:6;
  then
A3: PI+T(i) < r+PI/2;
  r+PI/2 < 3/2*PI+T(i)+PI/2 by A2,XREAL_1:6;
  then
A4: r+PI/2 < 2*PI+T(i);
  sin(r+PI/2) = cos r by SIN_COS:79;
  hence thesis by A3,A4,Th12;
end;
