reserve x for Real;

theorem
  rng cos = [.-1,1 .]
proof
  now
    let y be object;
    thus y in [.-1,1 .] implies ex x be object st x in dom cos & y = cos.x
    proof
      assume
A1:   y in [.-1,1 .];
      then reconsider y1=y as Real;
      cos|[.0,PI.] is continuous & y1 in [.cos.0,cos.PI.] \/ [.cos.PI,cos.
      0 .] by A1,SIN_COS:30,76,XBOOLE_0:def 3;
      then consider x be Real such that
      x in [.0,PI.] and
A2:   y1 = cos.x by FCONT_2:15,SIN_COS:24;
      take x;
       x in REAL by XREAL_0:def 1;
      hence thesis by A2,SIN_COS:24;
    end;
    thus (ex x be object st x in dom cos & y = cos.x) implies y in [.-1,1 .]
by
Th27,SIN_COS:24;
  end;
  hence thesis by FUNCT_1:def 3;
end;
