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

proof

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

end;

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