sin | ].(- (PI / 2)),(PI / 2).[ c= sin | [.(- (PI / 2)),(PI / 2).] by RELAT_1:104, XXREAL_1:25;
then A1: rng (sin | ].(- (PI / 2)),(PI / 2).[) c= rng (sin | [.(- (PI / 2)),(PI / 2).]) by RELAT_1:25;
A2: rng (sin | ].(- (PI / 2)),(PI / 2).[) = sin .: ].(- (PI / 2)),(PI / 2).[ by RELAT_1:148;
thus sin .: ].(- (PI / 2)),(PI / 2).[ c= ].(- 1),1.[ :: according to XBOOLE_0:def 10 :: thesis: ].(- 1),1.[ c= sin .: ].(- (PI / 2)),(PI / 2).[

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in sin .: ].(- (PI / 2)),(PI / 2).[ or x in ].(- 1),1.[ )
assume A3: x in sin .: ].(- (PI / 2)),(PI / 2).[ ; :: thesis: x in ].(- 1),1.[
then consider a being set such that
A4: a in dom sin and
A5: a in ].(- (PI / 2)),(PI / 2).[ and
A6: sin . a = x by FUNCT_1:def 12;
reconsider a = a, x = x as Real by A4, A6;
set i = [\(a / (2 * PI ))/];
A7: H₁([\(a / (2 * PI ))/]) / ((2 * PI ) * 1) = [\(a / (2 * PI ))/] / 1 by XCMPLX_1:92;
A8: sin . a = sin a by SIN_COS:def 21;

A9: now

assume x = 1 ; :: thesis: contradiction
then A10: a = (PI / 2) + H₁([\(a / (2 * PI ))/]) by A6, A8, Th24;
then (PI / 2) + H₁([\(a / (2 * PI ))/]) < PI / 2 by A5, XXREAL_1:4;
then ((PI / 2) + H₁([\(a / (2 * PI ))/])) - (PI / 2) < (PI / 2) - (PI / 2) by XREAL_1:11;
then [\(a / (2 * PI ))/] < 0 ;
then A11: [\(a / (2 * PI ))/] <= - 1 by INT_1:21;
- (PI / 2) < (PI / 2) + H₁([\(a / (2 * PI ))/]) by A5, A10, XXREAL_1:4;
then (- (PI / 2)) - (PI / 2) < ((PI / 2) + H₁([\(a / (2 * PI ))/])) - (PI / 2) by XREAL_1:11;
then ((- 1) * PI ) / (2 * PI ) < [\(a / (2 * PI ))/] by A7, XREAL_1:76;
then (- 1) / 2 < [\(a / (2 * PI ))/] by XCMPLX_1:92;
hence contradiction by A11, XXREAL_0:2; :: thesis: verum

end;

A12: now

assume x = - 1 ; :: thesis: contradiction
then A13: a = ((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/]) by A6, A8, Th23;
then - (PI / 2) < ((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/]) by A5, XXREAL_1:4;
then (- (PI / 2)) - ((3 / 2) * PI ) < (((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/])) - ((3 / 2) * PI ) by XREAL_1:11;
then (- (2 * PI )) / (2 * PI ) < [\(a / (2 * PI ))/] by A7, XREAL_1:76;
then - ((2 * PI ) / (2 * PI )) < [\(a / (2 * PI ))/] by XCMPLX_1:188;
then - 1 < [\(a / (2 * PI ))/] by XCMPLX_1:60;
then A14: (- 1) + 1 <= [\(a / (2 * PI ))/] by INT_1:20;
((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/]) < PI / 2 by A5, A13, XXREAL_1:4;
then (((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/])) - ((3 / 2) * PI ) < (PI / 2) - ((3 / 2) * PI ) by XREAL_1:11;
then [\(a / (2 * PI ))/] < (- PI ) / (2 * PI ) by A7, XREAL_1:76;
hence contradiction by A14, XCMPLX_1:92; :: thesis: verum

end;

x <= 1 by A1, A2, A3, COMPTRIG:48, XXREAL_1:1;
then A15: x < 1 by A9, XXREAL_0:1;
- 1 <= x by A1, A2, A3, COMPTRIG:48, XXREAL_1:1;
then - 1 < x by A12, XXREAL_0:1;
hence x in ].(- 1),1.[ by A15, XXREAL_1:4; :: thesis: verum

end;

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in ].(- 1),1.[ or a in sin .: ].(- (PI / 2)),(PI / 2).[ )
assume A16: a in ].(- 1),1.[ ; :: thesis: a in sin .: ].(- (PI / 2)),(PI / 2).[
then reconsider a = a as Real ;
( - 1 < a & a < 1 ) by A16, XXREAL_1:4;
then a in rng (sin | [.(- (PI / 2)),(PI / 2).]) by COMPTRIG:48, XXREAL_1:1;
then consider x being set such that
A17: x in dom (sin | [.(- (PI / 2)),(PI / 2).]) and
A18: (sin | [.(- (PI / 2)),(PI / 2).]) . x = a by FUNCT_1:def 5;
reconsider x = x as Real by A17;
A19: sin . x = a by A17, A18, FUNCT_1:70;
dom (sin | [.(- (PI / 2)),(PI / 2).]) = [.(- (PI / 2)),(PI / 2).] by RELAT_1:91, SIN_COS:27;
then ( - (PI / 2) <= x & x <= PI / 2 ) by A17, XXREAL_1:1;
then ( ( - (PI / 2) < x & x < PI / 2 ) or - (PI / 2) = x or PI / 2 = x ) by XXREAL_0:1;
then x in ].(- (PI / 2)),(PI / 2).[ by A16, A19, Th7, SIN_COS:81, XXREAL_1:4;
hence a in sin .: ].(- (PI / 2)),(PI / 2).[ by A19, FUNCT_1:def 12, SIN_COS:27; :: thesis: verum