reserve x for Real;

theorem Th27:
  sin.x in [.-1,1 .] & cos.x in [.-1,1 .]
proof
  |.cos x.| <= 1 by SIN_COS:27;
  then |.cos.x.| <= 1 by SIN_COS:def 19;
  then
A1: -1 <= cos.x & cos.x <= 1 by ABSVALUE:5;
  |.sin x.| <= 1 by SIN_COS:27;
  then |.sin.x.| <= 1 by SIN_COS:def 17;
  then -1 <= sin.x & sin.x <= 1 by ABSVALUE:5;
  hence thesis by A1,XXREAL_1:1;
end;
