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

now :: thesis: for y being object holds
( ( y in [.(- 1),1.] implies ex x being object st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) ) & ( ex x being object st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) implies y in [.(- 1),1.] ) )

let y be object ; :: thesis: ( ( y in [.(- 1),1.] implies ex x being object st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) ) & ( ex x being object 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 object st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) ) :: thesis: ( ex x being object 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 object st
( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x )
then reconsider y1 = y as Real ;
PI / 2 in [.(- (PI / 2)),(PI / 2).] by XXREAL_1:1;
then A2: (sin | [.(- (PI / 2)),(PI / 2).]) . (PI / 2) = sin . (PI / 2) by FUNCT_1:49;
- (PI / 2) in [.(- (PI / 2)),(PI / 2).] by XXREAL_1:1;
then (sin | [.(- (PI / 2)),(PI / 2).]) . (- (PI / 2)) = sin . (- (PI / 2)) by FUNCT_1:49;
then y1 in [.((sin | [.(- (PI / 2)),(PI / 2).]) . (- (PI / 2))),((sin | [.(- (PI / 2)),(PI / 2).]) . (PI / 2)).] by A1, A2, SIN_COS:30, SIN_COS:76;
then A3: ( (sin | [.(- (PI / 2)),(PI / 2).]) | [.(- (PI / 2)),(PI / 2).] is continuous & 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;
dom (sin | [.(- (PI / 2)),(PI / 2).]) = [.(- (PI / 2)),(PI / 2).] /\ REAL by RELAT_1:61, SIN_COS:24
.= [.(- (PI / 2)),(PI / 2).] by XBOOLE_1:28 ;
then consider x being Real such that
A4: x in [.(- (PI / 2)),(PI / 2).] and
A5: y1 = (sin | [.(- (PI / 2)),(PI / 2).]) . x by A3, FCONT_2:15;
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 A4, XBOOLE_0:def 4;
hence ( x in dom (sin | [.(- (PI / 2)),(PI / 2).]) & y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ) by A5, RELAT_1:61, SIN_COS:24; :: thesis: verum

end;

thus ( ex x being object 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 object such that A6: x in dom (sin | [.(- (PI / 2)),(PI / 2).]) and
A7: y = (sin | [.(- (PI / 2)),(PI / 2).]) . x ; :: thesis: y in [.(- 1),1.]
dom (sin | [.(- (PI / 2)),(PI / 2).]) c= dom sin by RELAT_1:60;
then reconsider x1 = x as Real by A6, SIN_COS:24;
y = sin . x1 by A6, A7, FUNCT_1:47;
hence y in [.(- 1),1.] by Th27; :: thesis: verum

end;

end;

hence rng (sin | [.(- (PI / 2)),(PI / 2).]) = [.(- 1),1.] by FUNCT_1:def 3; :: thesis: verum