set cos1 = cos | [.0,PI.];

now :: thesis: for y being object holds
( ( y in [.(- 1),1.] implies ex x being object st
( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x ) ) & ( ex x being object st
( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x ) implies y in [.(- 1),1.] ) )

let y be object ; :: thesis: ( ( y in [.(- 1),1.] implies ex x being object st
( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x ) ) & ( ex x being object st
( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x ) implies y in [.(- 1),1.] ) )
thus ( y in [.(- 1),1.] implies ex x being object st
( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x ) ) :: thesis: ( ex x being object st
( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x ) implies y in [.(- 1),1.] )

proof

PI in [.0,PI.] by XXREAL_1:1;
then A1: (cos | [.0,PI.]) . PI = cos . PI by FUNCT_1:49;
assume A2: y in [.(- 1),1.] ; :: thesis: ex x being object st
( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x )
then reconsider y1 = y as Real ;
A3: dom (cos | [.0,PI.]) = [.0,PI.] /\ REAL by RELAT_1:61, SIN_COS:24
.= [.0,PI.] by XBOOLE_1:28 ;
0 in [.0,PI.] by XXREAL_1:1;
then (cos | [.0,PI.]) . 0 = cos . 0 by FUNCT_1:49;
then ( (cos | [.0,PI.]) | [.0,PI.] is continuous & y1 in [.((cos | [.0,PI.]) . 0),((cos | [.0,PI.]) . PI).] \/ [.((cos | [.0,PI.]) . PI),((cos | [.0,PI.]) . 0).] ) by A2, A1, SIN_COS:30, SIN_COS:76, XBOOLE_0:def 3;
then consider x being Real such that
A4: x in [.0,PI.] and
A5: y1 = (cos | [.0,PI.]) . x by A3, FCONT_2:15;
take x ; :: thesis: ( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x )
x in REAL /\ [.0,PI.] by A4, XBOOLE_0:def 4;
hence ( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x ) by A5, RELAT_1:61, SIN_COS:24; :: thesis: verum

end;

thus ( ex x being object st
( x in dom (cos | [.0,PI.]) & y = (cos | [.0,PI.]) . x ) implies y in [.(- 1),1.] ) :: thesis: verum

proof

given x being object such that A6: x in dom (cos | [.0,PI.]) and
A7: y = (cos | [.0,PI.]) . x ; :: thesis: y in [.(- 1),1.]
dom (cos | [.0,PI.]) c= dom cos by RELAT_1:60;
then reconsider x1 = x as Real by A6, SIN_COS:24;
y = cos . x1 by A6, A7, FUNCT_1:47;
hence y in [.(- 1),1.] by Th27; :: thesis: verum

end;

end;

hence rng (cos | [.0,PI.]) = [.(- 1),1.] by FUNCT_1:def 3; :: thesis: verum