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

theorem Th12:
  PI+2*PI*i < r & r < 2*PI+2*PI*i implies sin r < 0
proof
  assume PI+T(i) < r & r < 2*PI+T(i);
  then PI+T(i)-T(i) < r-T(i) & r-T(i) < 2*PI+T(i)-T(i) by XREAL_1:9;
  then r-T(i) in ].PI,2*PI.[ by XXREAL_1:4;
  then sin.(r+T(-i)) < 0 by COMPTRIG:9;
  then sin.r < 0 by Th8;
  hence thesis by SIN_COS:def 17;
end;
