cos | ].0,PI.[ c= cos | [.0,PI.] by RELAT_1:75, XXREAL_1:25;
then A1: rng (cos | ].0,PI.[) c= rng (cos | [.0,PI.]) by RELAT_1:11;
A2: rng (cos | ].0,PI.[) = cos .: ].0,PI.[ by RELAT_1:115;
thus cos .: ].0,PI.[ c= ].(- 1),1.[ :: according to XBOOLE_0:def 10 :: thesis: ].(- 1),1.[ c= cos .: ].0,PI.[

proof

let x be object ; :: 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 object such that
A4: a in dom cos and
A5: a in ].0,PI.[ and
A6: cos . a = x by FUNCT_1:def 6;
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_1:91;
A8: cos . a = cos a by SIN_COS:def 19;

A9: now :: thesis: not x = 1

assume x = 1 ; :: thesis: contradiction
then A10: a = H₁([\(a / (2 * PI))/]) by A6, A8, Th26;
then H₁([\(a / (2 * PI))/]) < PI by A5, XXREAL_1:4;
then [\(a / (2 * PI))/] < (1 * PI) / (2 * PI) by A7, XREAL_1:74;
then A11: [\(a / (2 * PI))/] < 1 / 2 by XCMPLX_1:91;
0 < [\(a / (2 * PI))/] by A5, A10, XXREAL_1:4;
then 0 + 1 <= [\(a / (2 * PI))/] by INT_1:7;
hence contradiction by A11, XXREAL_0:2; :: thesis: verum

end;

A12: now :: thesis: not x = - 1

assume x = - 1 ; :: thesis: contradiction
then A13: a = PI + H₁([\(a / (2 * PI))/]) by A6, A8, Th25;
then 0 < PI + H₁([\(a / (2 * PI))/]) by A5, XXREAL_1:4;
then 0 - PI < (PI + H₁([\(a / (2 * PI))/])) - PI by XREAL_1:9;
then (- PI) / (2 * PI) < H₁([\(a / (2 * PI))/]) / (2 * PI) by XREAL_1:74;
then - ((1 * PI) / (2 * PI)) < [\(a / (2 * PI))/] by A7, XCMPLX_1:187;
then A14: - (1 / 2) < [\(a / (2 * PI))/] by XCMPLX_1:91;
PI + H₁([\(a / (2 * PI))/]) < PI by A5, A13, XXREAL_1:4;
then (PI + H₁([\(a / (2 * PI))/])) - PI < PI - PI by XREAL_1:9;
then [\(a / (2 * PI))/] < 0 ;
then [\(a / (2 * PI))/] <= - 1 by INT_1:8;
hence contradiction by A14, XXREAL_0:2; :: thesis: verum

end;

x <= 1 by A1, A2, A3, COMPTRIG:32, XXREAL_1:1;
then A15: x < 1 by A9, XXREAL_0:1;
- 1 <= x by A1, A2, A3, COMPTRIG:32, 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 object ; :: according to TARSKI:def 3 :: thesis: ( not a in ].(- 1),1.[ or a in cos .: ].0,PI.[ )
assume A16: a in ].(- 1),1.[ ; :: thesis: a in cos .: ].0,PI.[
then reconsider a = a as Real ;
( - 1 < a & a < 1 ) by A16, XXREAL_1:4;
then a in rng (cos | [.0,PI.]) by COMPTRIG:32, XXREAL_1:1;
then consider x being object such that
A17: x in dom (cos | [.0,PI.]) and
A18: (cos | [.0,PI.]) . x = a by FUNCT_1:def 3;
reconsider x = x as Real by A17;
A19: cos . x = a by A17, A18, FUNCT_1:47;
A20: dom (cos | [.0,PI.]) = [.0,PI.] by RELAT_1:62, SIN_COS:24;
then x <= PI by A17, XXREAL_1:1;
then ( ( 0 < x & x < PI ) or 0 = x or PI = x ) by A17, A20, XXREAL_0:1, XXREAL_1:1;
then x in ].0,PI.[ by A16, A19, SIN_COS:30, SIN_COS:76, XXREAL_1:4;
hence a in cos .: ].0,PI.[ by A19, FUNCT_1:def 6, SIN_COS:24; :: thesis: verum