A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def 12;
rng (tan | [.(- (PI / 4)),(PI / 4).]) c= rng (tan | ].(- (PI / 2)),(PI / 2).[)

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (tan | [.(- (PI / 4)),(PI / 4).]) or y in rng (tan | ].(- (PI / 2)),(PI / 2).[) )
assume y in rng (tan | [.(- (PI / 4)),(PI / 4).]) ; :: thesis: y in rng (tan | ].(- (PI / 2)),(PI / 2).[)
then y in tan .: [.(- (PI / 4)),(PI / 4).] by RELAT_1:115;
then ex x being object st
( x in dom tan & x in [.(- (PI / 4)),(PI / 4).] & y = tan . x ) by FUNCT_1:def 6;
then y in tan .: ].(- (PI / 2)),(PI / 2).[ by A1, FUNCT_1:def 6;
hence y in rng (tan | ].(- (PI / 2)),(PI / 2).[) by RELAT_1:115; :: thesis: verum

end;

hence [.(- 1),1.] c= dom arctan by Th21, FUNCT_1:33; :: thesis: verum