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

theorem
  PI+2*PI*i <= r & r <= 2*PI+2*PI*i implies sin r <= 0
proof
  assume PI+2*PI*i <= r & r <= 2*PI+2*PI*i;
  then PI+2*PI*i < r & r < 2*PI+2*PI*i or PI+2*PI*i = r or r = 2*PI+2*PI*i by
XXREAL_0:1;
  hence thesis by Th12,COMPLEX2:8,SIN_COS:77;
end;
