cos | ].PI ,(2 * PI ).[ c= cos | [.PI ,(2 * PI ).] by Lm22, RELAT_1:104;
then A1: rng (cos | ].PI ,(2 * PI ).[) c= rng (cos | [.PI ,(2 * PI ).]) by RELAT_1:25;
A2: rng (cos | ].PI ,(2 * PI ).[) = cos .: ].PI ,(2 * PI ).[ by RELAT_1:148;
thus cos .: ].PI ,(2 * PI ).[ c= ].(- 1),1.[ :: according to XBOOLE_0:def 10 :: thesis: ].(- 1),1.[ c= cos .: ].PI ,(2 * PI ).[

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in cos .: ].PI ,(2 * PI ).[ or x in ].(- 1),1.[ )
assume A3: x in cos .: ].PI ,(2 * PI ).[ ; :: thesis: x in ].(- 1),1.[
then consider a being set such that
A4: a in dom cos and
A5: a in ].PI ,(2 * PI ).[ and
A6: cos . a = x by FUNCT_1:def 12;
reconsider a = a, x = x as Real by A4, A6;
A7: cos . a = cos a by SIN_COS:def 23;
A8: ( - 1 <= x & x <= 1 ) by A1, A2, A3, COMPTRIG:51, 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 = H₁([\(a / (2 * PI ))/]) by A6, A7, Th26;
then ( PI < H₁([\(a / (2 * PI ))/]) & H₁([\(a / (2 * PI ))/]) < 2 * PI ) by A5, XXREAL_1:4;
then ( (1 * PI ) / (2 * PI ) < [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] < (2 * PI ) / (2 * PI ) ) by A9, XREAL_1:76;
then ( 1 / 2 < [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] < 1 + 0 ) by XCMPLX_1:60, XCMPLX_1:92;
hence contradiction by INT_1:20; :: thesis: verum

end;

now

assume x = - 1 ; :: thesis: contradiction
then a = PI + H₁([\(a / (2 * PI ))/]) by A6, A7, Th25;
then ( PI < PI + H₁([\(a / (2 * PI ))/]) & PI + H₁([\(a / (2 * PI ))/]) < 2 * PI ) by A5, XXREAL_1:4;
then ( PI - PI < (PI + H₁([\(a / (2 * PI ))/])) - PI & (PI + H₁([\(a / (2 * PI ))/])) - PI < (2 * PI ) - PI ) by XREAL_1:11;
then ( 0 / (2 * PI ) < [\(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;

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 cos .: ].PI ,(2 * PI ).[ )
assume A11: a in ].(- 1),1.[ ; :: thesis: a in cos .: ].PI ,(2 * PI ).[
then reconsider a = a as Real ;
( - 1 < a & a < 1 ) by A11, XXREAL_1:4;
then a in rng (cos | [.PI ,(2 * PI ).]) by COMPTRIG:51, XXREAL_1:1;
then consider x being set such that
A12: x in dom (cos | [.PI ,(2 * PI ).]) and
A13: (cos | [.PI ,(2 * PI ).]) . x = a by FUNCT_1:def 5;
A14: dom (cos | [.PI ,(2 * PI ).]) = [.PI ,(2 * PI ).] by Lm2, RELAT_1:91;
reconsider x = x as Real by A12;
A15: cos . x = a by A12, A13, FUNCT_1:70;
( PI <= x & x <= 2 * PI ) by A12, A14, XXREAL_1:1;
then ( ( PI < x & x < 2 * PI ) or PI = x or 2 * PI = x ) by XXREAL_0:1;
then x in ].PI ,(2 * PI ).[ by A11, A15, SIN_COS:81, XXREAL_1:4;
hence a in cos .: ].PI ,(2 * PI ).[ by A15, Lm2, FUNCT_1:def 12; :: thesis: verum