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

now

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

proof

assume A1: y in [.(- 1),1.] ; :: thesis: ex x being set st
( x in dom (cos | [.0 ,PI .]) & y = (cos | [.0 ,PI .]) . x )
then reconsider y1 = y as Real ;
A2: (cos | [.0 ,PI .]) | [.0 ,PI .] is continuous ;
X: dom (cos | [.0 ,PI .]) = [.0 ,PI .] /\ REAL by RELAT_1:90, SIN_COS:27
.= [.0 ,PI .] by XBOOLE_1:28 ;
( 0 in [.0 ,PI .] & PI in [.0 ,PI .] ) by XXREAL_1:1;
then ( (cos | [.0 ,PI .]) . 0 = cos . 0 & (cos | [.0 ,PI .]) . PI = cos . PI ) by FUNCT_1:72;
then y1 in [.((cos | [.0 ,PI .]) . 0 ),((cos | [.0 ,PI .]) . PI ).] \/ [.((cos | [.0 ,PI .]) . PI ),((cos | [.0 ,PI .]) . 0 ).] by A1, SIN_COS:33, SIN_COS:81, XBOOLE_0:def 3;
then consider x being Real such that
A3: x in [.0 ,PI .] and
A4: y1 = (cos | [.0 ,PI .]) . x by A2, X, FCONT_2:16;
take x ; :: thesis: ( x in dom (cos | [.0 ,PI .]) & y = (cos | [.0 ,PI .]) . x )
x in REAL /\ [.0 ,PI .] by A3, XBOOLE_0:def 4;
hence ( x in dom (cos | [.0 ,PI .]) & y = (cos | [.0 ,PI .]) . x ) by A4, RELAT_1:90, SIN_COS:27; :: thesis: verum

end;

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

proof

given x being set such that A5: x in dom (cos | [.0 ,PI .]) and
A6: y = (cos | [.0 ,PI .]) . x ; :: thesis: y in [.(- 1),1.]
dom (cos | [.0 ,PI .]) c= dom cos by RELAT_1:89;
then reconsider x1 = x as Real by A5, SIN_COS:27;
y = cos . x1 by A5, A6, FUNCT_1:70;
hence y in [.(- 1),1.] by Th45; :: thesis: verum

end;

end;

hence rng (cos | [.0 ,PI .]) = [.(- 1),1.] by FUNCT_1:def 5; :: thesis: verum