let r be Real; :: thesis: ( - 1 <= r & r <= 1 implies ( - (PI / 4) <= arctan r & arctan r <= PI / 4 ) )
assume ( - 1 <= r & r <= 1 ) ; :: thesis: ( - (PI / 4) <= arctan r & arctan r <= PI / 4 )
then A1: r in [.(- 1),1.] by XXREAL_1:1;
then r in dom (arctan | [.(- 1),1.]) by Th21, RELAT_1:91;
then (arctan | [.(- 1),1.]) . r in rng (arctan | [.(- 1),1.]) by FUNCT_1:def 5;
then arctan r in rng (arctan | [.(- 1),1.]) by A1, FUNCT_1:72;
hence ( - (PI / 4) <= arctan r & arctan r <= PI / 4 ) by Th53, XXREAL_1:1; :: thesis: verum