PI in ].0 ,4.[ by SIN_COS:def 32;
then A1: 0 < PI by XXREAL_1:4;
then 0 / 2 < PI / 2 by XREAL_1:76;
then (PI / 2) * (- 1) < 0 * (- 1) by XREAL_1:71;
then A2: (- (PI / 2)) + ((2 * PI ) * 0 ) < PI / 4 by A1;
PI / 4 < (PI / 2) + ((2 * PI ) * 0 ) by A1, XREAL_1:78;
hence cos (PI / 4) > 0 by A2, SIN_COS6:13; :: thesis: verum