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

theorem
  3/2*PI+2*PI*i <= r & r <= 2*PI+2*PI*i implies cos r >= 0
proof
  assume
A1: 3/2*PI+2*PI*i <= r & r <= 2*PI+2*PI*i;
  per cases by A1,XXREAL_0:1;
  suppose
    3/2*PI+2*PI*i < r & r < 2*PI+2*PI*i;
    hence thesis by Th15;
  end;
  suppose
    3/2*PI+2*PI*i = r;
    hence thesis by COMPLEX2:9,SIN_COS:77;
  end;
  suppose
    r = 2*PI+2*PI*i;
    hence thesis by COMPLEX2:9,SIN_COS:77;
  end;
end;
