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

proof

let x be real number ; :: thesis: ( x in dom (cos ^2 ) implies ( x + PI in dom (cos ^2 ) & x - PI in dom (cos ^2 ) & (cos ^2 ) . x = (cos ^2 ) . (x + PI ) ) )
assume x in dom (cos ^2 ) ; :: thesis: ( x + PI in dom (cos ^2 ) & x - PI in dom (cos ^2 ) & (cos ^2 ) . x = (cos ^2 ) . (x + PI ) )
A2: ( x + PI in dom cos & x - PI in dom cos ) by SIN_COS:27;
(cos ^2 ) . (x + PI ) = (cos (x + PI )) ^2 by VALUED_1:11
.= (- (cos . x)) ^2 by SIN_COS:83
.= (cos . x) ^2
.= (cos ^2 ) . x by VALUED_1:11 ;
hence ( x + PI in dom (cos ^2 ) & x - PI in dom (cos ^2 ) & (cos ^2 ) . x = (cos ^2 ) . (x + PI ) ) by A2, VALUED_1:11; :: thesis: verum

end;

hence cos ^2 is PI -periodic by Th1; :: thesis: verum