set sin1 = sin | [.(- (PI / 2)),(PI / 2).];

now

let y be set ; :: thesis: ( ( y in [.(- 1),1.] implies ex x being set st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) ) & ( ex x being set st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) implies y in [.(- 1),1.] ) )
thus ( y in [.(- 1),1.] implies ex x being set st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) ) :: thesis: ( ex x being set st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) implies y in [.(- 1),1.] )

proof

assume A1: y in [.(- 1),1.] ; :: thesis: ex x being set st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x )
then reconsider y1 = y as Real ;
A2: (sin | [.(- (PI / 2)),(PI / 2).]) | [.(- (PI / 2)),(PI / 2).] is continuous ;
X: dom (sin | [.(- (PI / 2)),(PI / 2).]) = [.(- (PI / 2)),(PI / 2).] /\ REAL by RELAT_1:90, SIN_COS:27
.= [.(- (PI / 2)),(PI / 2).] by XBOOLE_1:28 ;
( - (PI / 2) in [.(- (PI / 2)),(PI / 2).] & PI / 2 in [.(- (PI / 2)),(PI / 2).] ) by XXREAL_1:1;
then ( (sin | [.(- (PI / 2)),(PI / 2).]) . (PI / 2) = sin . (PI / 2) & (sin | [.(- (PI / 2)),(PI / 2).]) . (- (PI / 2)) = sin . (- (PI / 2)) ) by FUNCT_1:72;
then y1 in [.((sin | [.(- (PI / 2)),(PI / 2).]) . (- (PI / 2))),((sin | [.(- (PI / 2)),(PI / 2).]) . (PI / 2)).] by A1, SIN_COS:33, SIN_COS:81;
then y1 in [.((sin | [.(- (PI / 2)),(PI / 2).]) . (- (PI / 2))),((sin | [.(- (PI / 2)),(PI / 2).]) . (PI / 2)).] \/ [.((sin | [.(- (PI / 2)),(PI / 2).]) . (PI / 2)),((sin | [.(- (PI / 2)),(PI / 2).]) . (- (PI / 2))).] by XBOOLE_0:def 3;
then consider x being Real such that
A3: x in [.(- (PI / 2)),(PI / 2).] and
A4: y1 = (sin | [.(- (PI / 2)),(PI / 2).]) . x by A2, X, FCONT_2:16;
take x ; :: thesis: ( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x )
x in REAL /\ [.(- (PI / 2)),(PI / 2).] by A3, XBOOLE_0:def 4;
hence ( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) by A4, RELAT_1:90, SIN_COS:27; :: thesis: verum

end;

thus ( ex x being set st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) implies y in [.(- 1),1.] ) :: thesis: verum

proof

given x being set such that A5: x in dom (sin | [.(- (PI / 2)),(PI / 2).]) and
A6: y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ; :: thesis: y in [.(- 1),1.]
dom (sin | [.(- (PI / 2)),(PI / 2).]) c= dom sin by RELAT_1:89;
then reconsider x1 = x as Real by A5, SIN_COS:27;
y = sin . x1 by A5, A6, FUNCT_1:70;
hence y in [.(- 1),1.] by Th45; :: thesis: verum

end;

end;

hence rng (sin | [.(- (PI / 2)),(PI / 2).]) = [.(- 1),1.] by FUNCT_1:def 5; :: thesis: verum