reserve x for Real;

theorem
  rng sin = [.-1,1 .]
proof
  now
    let y be object;
    thus y in [.-1,1 .] implies ex x be object st x in dom sin & y = sin.x
    proof
      assume
A1:   y in [.-1,1 .];
      then reconsider y1=y as Real;
      y1 in [.-1,1 .] \/ [.1,sin.(-PI/2).] by A1,XBOOLE_0:def 3;
      then
      sin|[.-PI/2,PI/2.] is continuous & y1 in [.sin.(-PI/2),sin.(PI/2).]
      \/ [.sin .(PI/2),sin.(-PI/2).] by SIN_COS:30,76;
      then consider x be Real such that
      x in [.-PI/2,PI/2 .] and
A2:   y1 = sin.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 sin & y = sin.x) implies y in [.-1,1 .]
    by Th27,SIN_COS:24;
  end;
  hence thesis by FUNCT_1:def 3;
end;
