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

proof

let x be real number ; :: thesis: ( x in dom cot implies ( x + PI in dom cot & x - PI in dom cot & cot . x = cot . (x + PI ) ) )
assume A1: x in dom cot ; :: thesis: ( x + PI in dom cot & x - PI in dom cot & cot . x = cot . (x + PI ) )
then x in (dom cos ) /\ ((dom sin ) \ (sin " {0 })) by RFUNCT_1:def 4;
then ( x in dom cos & x in (dom sin ) \ (sin " {0 }) ) by XBOOLE_0:def 4;
then ( x in dom cos & x in dom sin & not x in sin " {0 } ) by XBOOLE_0:def 5;
then not sin . x in {0 } by FUNCT_1:def 13;
then B1: sin . x <> 0 by TARSKI:def 1;
B2: sin . (x + PI ) = - (sin . x) by SIN_COS:83;
then not sin . (x + PI ) in {0 } by TARSKI:def 1, B1;
then ( x + PI in dom cos & x + PI in dom sin & not x + PI in sin " {0 } ) by FUNCT_1:def 13, SIN_COS:27;
then ( x + PI in dom cos & x + PI in (dom sin ) \ (sin " {0 }) ) by XBOOLE_0:def 5;
then A3: x + PI in (dom cos ) /\ ((dom sin ) \ (sin " {0 })) by XBOOLE_0:def 4;
sin . (x - PI ) = sin . ((x - PI ) + (2 * PI )) by SIN_COS:83
.= sin . (x + PI ) ;
then not sin . (x - PI ) in {0 } by B1, B2, TARSKI:def 1;
then ( x - PI in dom cos & x - PI in dom sin & not x - PI in sin " {0 } ) by FUNCT_1:def 13, SIN_COS:27;
then ( x - PI in dom cos & x - PI in (dom sin ) \ (sin " {0 }) ) by XBOOLE_0:def 5;
then A4: x - PI in (dom cos ) /\ ((dom sin ) \ (sin " {0 })) by XBOOLE_0:def 4;
then A5: ( x + PI in dom cot & x - PI in dom cot ) by A3, RFUNCT_1:def 4;
cot . (x + PI ) = (cos . (x + PI )) / (sin . (x + PI )) by A5, RFUNCT_1:def 4
.= (- (cos . x)) / (sin . (x + PI )) by SIN_COS:83
.= (- (cos . x)) / (- (sin . x)) by SIN_COS:83
.= (cos . x) / (sin . x) by XCMPLX_1:192
.= cot . x by A1, RFUNCT_1:def 4 ;
hence ( x + PI in dom cot & x - PI in dom cot & cot . x = cot . (x + PI ) ) by A3, A4, RFUNCT_1:def 4; :: thesis: verum

end;

hence cot is PI -periodic by Th1; :: thesis: verum