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

PI in [.0 ,PI .] by XXREAL_1:1;
then A1: (cos | [.0 ,PI .]) . PI = cos . PI by FUNCT_1:72;
assume A2: 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 ;
A3: dom (cos | [.0 ,PI .]) = [.0 ,PI .] /\ REAL by RELAT_1:90, SIN_COS:27
.= [.0 ,PI .] by XBOOLE_1:28 ;
0 in [.0 ,PI .] by XXREAL_1:1;
then (cos | [.0 ,PI .]) . 0 = cos . 0 by FUNCT_1:72;
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:33, SIN_COS:81, 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_2:16;
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_1:90, 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 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_1:89;
then reconsider x1 = x as Real by A6, SIN_COS:27;
y = cos . x1 by A6, A7, 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