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

theorem
  2*PI*i <= r & r < PI+2*PI*i implies cos r > -1
proof
  assume that
A1: T(i) <= r & r < PI+T(i) and
A2: cos r <= -1;
A3: T(i)-T(i) <= r-T(i) & r-T(i) < PI+T(i)-T(i) by A1,XREAL_1:9;
A4: cos r >= -1 by Th5;
  cos(r-T(i)) = cos(r+T(-i)) .= cos r by COMPLEX2:9
    .= -1 by A2,A4,XXREAL_0:1;
  hence thesis by A3,Th35;
end;
