sin | ].(PI / 2),((3 / 2) * PI ).[ c= sin | [.(PI / 2),((3 / 2) * PI ).] by Lm20, RELAT_1:104;
then A1: rng (sin | ].(PI / 2),((3 / 2) * PI ).[) c= rng (sin | [.(PI / 2),((3 / 2) * PI ).]) by RELAT_1:25;
A2: rng (sin | ].(PI / 2),((3 / 2) * PI ).[) = sin .: ].(PI / 2),((3 / 2) * PI ).[ by RELAT_1:148;
thus sin .: ].(PI / 2),((3 / 2) * PI ).[ c= ].(- 1),1.[ :: according to XBOOLE_0:def 10 :: thesis: ].(- 1),1.[ c= sin .: ].(PI / 2),((3 / 2) * PI ).[

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in sin .: ].(PI / 2),((3 / 2) * PI ).[ or x in ].(- 1),1.[ )
assume A3: x in sin .: ].(PI / 2),((3 / 2) * PI ).[ ; :: thesis: x in ].(- 1),1.[
then consider a being set such that
A4: a in dom sin and
A5: a in ].(PI / 2),((3 / 2) * PI ).[ and
A6: sin . a = x by FUNCT_1:def 12;
reconsider a = a, x = x as Real by A4, A6;
A7: sin . a = sin a by SIN_COS:def 21;
A8: ( - 1 <= x & x <= 1 ) by A1, A2, A3, COMPTRIG:49, XXREAL_1:1;
set i = [\(a / (2 * PI ))/];
A9: H₁([\(a / (2 * PI ))/]) / ((2 * PI ) * 1) = [\(a / (2 * PI ))/] / 1 by XCMPLX_1:92;

A10: now

assume x = 1 ; :: thesis: contradiction
then a = (PI / 2) + H₁([\(a / (2 * PI ))/]) by A6, A7, Th24;
then ( PI / 2 < (PI / 2) + H₁([\(a / (2 * PI ))/]) & (PI / 2) + H₁([\(a / (2 * PI ))/]) < (3 / 2) * PI ) by A5, XXREAL_1:4;
then ( (PI / 2) - (PI / 2) < ((PI / 2) + H₁([\(a / (2 * PI ))/])) - (PI / 2) & ((PI / 2) + H₁([\(a / (2 * PI ))/])) - (PI / 2) < ((3 / 2) * PI ) - (PI / 2) ) by XREAL_1:11;
then ( 0 / (2 * PI ) < [\(a / (2 * PI ))/] & H₁([\(a / (2 * PI ))/]) < PI ) ;
then ( 0 < [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] < (1 * PI ) / (2 * PI ) ) by A9, XREAL_1:76;
then ( 0 + 1 <= [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] < 1 / 2 ) by INT_1:20, XCMPLX_1:92;
hence contradiction by XXREAL_0:2; :: thesis: verum

end;

now

assume x = - 1 ; :: thesis: contradiction
then a = ((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/]) by A6, A7, Th23;
then ( PI / 2 < ((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/]) & ((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/]) < (3 / 2) * PI ) by A5, XXREAL_1:4;
then ( (PI / 2) - ((3 / 2) * PI ) < (((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/])) - ((3 / 2) * PI ) & (((3 / 2) * PI ) + H₁([\(a / (2 * PI ))/])) - ((3 / 2) * PI ) < ((3 / 2) * PI ) - ((3 / 2) * PI ) ) by XREAL_1:11;
then ( (- PI ) / (2 * PI ) < H₁([\(a / (2 * PI ))/]) / (2 * PI ) & H₁([\(a / (2 * PI ))/]) / (2 * PI ) < 0 / (2 * PI ) ) by XREAL_1:76;
then ( ((- 1) * PI ) / (2 * PI ) < [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] < 0 ) by A9;
then ( (- 1) / 2 < [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] < 0 ) by XCMPLX_1:92;
then ( - (1 / 2) < [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] <= - 1 ) by INT_1:21;
hence contradiction by XXREAL_0:2; :: thesis: verum

end;

then ( - 1 < x & x < 1 ) by A8, A10, XXREAL_0:1;
hence x in ].(- 1),1.[ by 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),((3 / 2) * PI ).[ )
assume A11: a in ].(- 1),1.[ ; :: thesis: a in sin .: ].(PI / 2),((3 / 2) * PI ).[
then reconsider a = a as Real ;
( - 1 < a & a < 1 ) by A11, XXREAL_1:4;
then a in rng (sin | [.(PI / 2),((3 / 2) * PI ).]) by COMPTRIG:49, XXREAL_1:1;
then consider x being set such that
A12: x in dom (sin | [.(PI / 2),((3 / 2) * PI ).]) and
A13: (sin | [.(PI / 2),((3 / 2) * PI ).]) . x = a by FUNCT_1:def 5;
A14: dom (sin | [.(PI / 2),((3 / 2) * PI ).]) = [.(PI / 2),((3 / 2) * PI ).] by Lm1, RELAT_1:91;
reconsider x = x as Real by A12;
A15: sin . x = a by A12, A13, FUNCT_1:70;
( PI / 2 <= x & x <= (3 / 2) * PI ) by A12, A14, XXREAL_1:1;
then ( ( PI / 2 < x & x < (3 / 2) * PI ) or PI / 2 = x or (3 / 2) * PI = x ) by XXREAL_0:1;
then x in ].(PI / 2),((3 / 2) * PI ).[ by A11, A15, SIN_COS:81, XXREAL_1:4;
hence a in sin .: ].(PI / 2),((3 / 2) * PI ).[ by A15, Lm1, FUNCT_1:def 12; :: thesis: verum