for x being real number st x in dom sec holds
( x + (2 * PI ) in dom sec & x - (2 * PI ) in dom sec & sec . x = sec . (x + (2 * PI )) )

proof

let x be real number ; :: thesis: ( x in dom sec implies ( x + (2 * PI ) in dom sec & x - (2 * PI ) in dom sec & sec . x = sec . (x + (2 * PI )) ) )
assume A1: x in dom sec ; :: thesis: ( x + (2 * PI ) in dom sec & x - (2 * PI ) in dom sec & sec . x = sec . (x + (2 * PI )) )
then x in (dom cos ) \ (cos " {0 }) by RFUNCT_1:def 8;
then ( x in dom cos & not x in cos " {0 } ) by XBOOLE_0:def 5;
then A2: not cos . x in {0 } by FUNCT_1:def 13;
then cos . x <> 0 by TARSKI:def 1;
then cos . (x + (2 * PI )) <> 0 by SIN_COS:83;
then not cos . (x + (2 * PI )) in {0 } by TARSKI:def 1;
then ( x + (2 * PI ) in dom cos & not x + (2 * PI ) in cos " {0 } ) by FUNCT_1:def 13, SIN_COS:27;
then A3: x + (2 * PI ) in (dom cos ) \ (cos " {0 }) by XBOOLE_0:def 5;
cos . (x - (2 * PI )) = cos . ((x - (2 * PI )) + (2 * PI )) by L2, Th1, SIN_COS:27;
then ( x - (2 * PI ) in dom cos & not x - (2 * PI ) in cos " {0 } ) by A2, FUNCT_1:def 13, SIN_COS:27;
then A4: x - (2 * PI ) in (dom cos ) \ (cos " {0 }) by XBOOLE_0:def 5;
then A5: ( x + (2 * PI ) in dom sec & x - (2 * PI ) in dom sec ) by A3, RFUNCT_1:def 8;
sec . (x + (2 * PI )) = (cos . (x + (2 * PI ))) " by A5, RFUNCT_1:def 8
.= (cos . x) " by SIN_COS:83
.= sec . x by A1, RFUNCT_1:def 8 ;
hence ( x + (2 * PI ) in dom sec & x - (2 * PI ) in dom sec & sec . x = sec . (x + (2 * PI )) ) by A3, A4, RFUNCT_1:def 8; :: thesis: verum

end;

hence sec is 2 * PI -periodic by Th1; :: thesis: verum