cos | ].0 ,PI .[ c= cos | [.0 ,PI .] by Lm18, RELAT_1:104;
then A1: rng (cos | ].0 ,PI .[) c= rng (cos | [.0 ,PI .]) by RELAT_1:25;
A2: rng (cos | ].0 ,PI .[) = cos .: ].0 ,PI .[ by RELAT_1:148;
thus cos .: ].0 ,PI .[ c= ].(- 1),1.[ :: according to XBOOLE_0:def 10 :: thesis: ].(- 1),1.[ c= cos .: ].0 ,PI .[

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in cos .: ].0 ,PI .[ or x in ].(- 1),1.[ )
assume A3: x in cos .: ].0 ,PI .[ ; :: thesis: x in ].(- 1),1.[
then consider a being set such that
A4: a in dom cos and
A5: a in ].0 ,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:50, 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 + H₁([\(a / (2 * PI ))/]) by A6, A7, Th25;
then ( 0 < PI + H₁([\(a / (2 * PI ))/]) & PI + H₁([\(a / (2 * PI ))/]) < PI ) by A5, XXREAL_1:4;
then ( 0 - PI < (PI + H₁([\(a / (2 * PI ))/])) - PI & (PI + H₁([\(a / (2 * PI ))/])) - PI < PI - 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, XCMPLX_1:188;
then ( - (1 / 2) < [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] <= - 1 ) by INT_1:21, XCMPLX_1:92;
hence contradiction by XXREAL_0:2; :: thesis: verum

end;

now

assume x = 1 ; :: thesis: contradiction
then a = H₁([\(a / (2 * PI ))/]) by A6, A7, Th26;
then ( 0 < H₁([\(a / (2 * PI ))/]) & H₁([\(a / (2 * PI ))/]) < PI ) by A5, XXREAL_1:4;
then ( 0 / (2 * PI ) < H₁([\(a / (2 * PI ))/]) / (2 * PI ) & H₁([\(a / (2 * PI ))/]) / (2 * PI ) < PI / (2 * PI ) ) by XREAL_1:76;
then ( 0 < [\(a / (2 * PI ))/] & [\(a / (2 * PI ))/] < (1 * PI ) / (2 * PI ) ) by A9;
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 .: ].0 ,PI .[ )
assume A11: a in ].(- 1),1.[ ; :: thesis: a in cos .: ].0 ,PI .[
then reconsider a = a as Real ;
( - 1 < a & a < 1 ) by A11, XXREAL_1:4;
then a in rng (cos | [.0 ,PI .]) by COMPTRIG:50, XXREAL_1:1;
then consider x being set such that
A12: x in dom (cos | [.0 ,PI .]) and
A13: (cos | [.0 ,PI .]) . x = a by FUNCT_1:def 5;
A14: dom (cos | [.0 ,PI .]) = [.0 ,PI .] by Lm2, RELAT_1:91;
reconsider x = x as Real by A12;
A15: cos . x = a by A12, A13, FUNCT_1:70;
( 0 <= x & x <= PI ) by A12, A14, XXREAL_1:1;
then ( ( 0 < x & x < PI ) or 0 = x or PI = x ) by XXREAL_0:1;
then x in ].0 ,PI .[ by A11, A15, SIN_COS:33, SIN_COS:81, XXREAL_1:4;
hence a in cos .: ].0 ,PI .[ by A15, Lm2, FUNCT_1:def 12; :: thesis: verum