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

theorem Th15:
  3/2*PI+2*PI*i < r & r < 2*PI+2*PI*i implies cos r > 0
proof
  assume that
A1: 3/2*PI+T(i) < r and
A2: r < 2*PI+T(i);
  3/2*PI+T(i)-PI < r-PI by A1,XREAL_1:9;
  then
A3: PI/2+T(i) < r-PI;
  PI+T(i) < 3/2*PI+T(i) & r-PI < 2*PI+T(i)-PI by A2,COMPTRIG:5,XREAL_1:6,9;
  then
A4: r-PI < 3/2*PI+T(i) by XXREAL_0:2;
  cos(r-PI) = cos(-(PI-r)) .= cos(PI+-r) by SIN_COS:31
    .= -cos(-r) by SIN_COS:79
    .= -cos r by SIN_COS:31;
  hence thesis by A3,A4,Th14;
end;
