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

theorem Th13:
  -PI/2+2*PI*i < r & r < PI/2+2*PI*i implies cos r > 0
proof
  assume that
A1: -PI/2+T(i) < r and
A2: r < PI/2+T(i);
  r+PI/2 < PI/2+T(i)+PI/2 by A2,XREAL_1:6;
  then
A3: r+PI/2 < PI+T(i);
A4: sin(r+PI/2) = cos r by SIN_COS:79;
  -PI/2+T(i)+PI/2 < r+PI/2 by A1,XREAL_1:6;
  hence thesis by A3,A4,Th11;
end;
