A1: [.(- (PI / 2)),(- (PI / 4)).] c= [.(- (PI / 2)),0 .[ by Lm7, XXREAL_2:def 12;
rng (cosec | [.(- (PI / 2)),(- (PI / 4)).]) c= rng (cosec | [.(- (PI / 2)),0 .[)

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (cosec | [.(- (PI / 2)),(- (PI / 4)).]) or y in rng (cosec | [.(- (PI / 2)),0 .[) )
assume y in rng (cosec | [.(- (PI / 2)),(- (PI / 4)).]) ; :: thesis: y in rng (cosec | [.(- (PI / 2)),0 .[)
then y in cosec .: [.(- (PI / 2)),(- (PI / 4)).] by RELAT_1:148;
then ex x being set st
( x in dom cosec & x in [.(- (PI / 2)),(- (PI / 4)).] & y = cosec . x ) by FUNCT_1:def 12;
then y in cosec .: [.(- (PI / 2)),0 .[ by A1, FUNCT_1:def 12;
hence y in rng (cosec | [.(- (PI / 2)),0 .[) by RELAT_1:148; :: thesis: verum

end;

hence [.(- (sqrt 2)),(- 1).] c= dom arccosec1 by Th43, FUNCT_1:55; :: thesis: verum