let k be Nat; :: thesis: cot is PI * (k + 1) -periodic
defpred S₁[ Nat] means cot is PI * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by L6;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: cot is PI * (k + 1) -periodic ; :: thesis: S₁[k + 1]
cot is PI * ((k + 1) + 1) -periodic

proof

for x being real number st x in dom cot holds
( x + (PI * ((k + 1) + 1)) in dom cot & x - (PI * ((k + 1) + 1)) in dom cot & cot . x = cot . (x + (PI * ((k + 1) + 1))) )

proof

let x be real number ; :: thesis: ( x in dom cot implies ( x + (PI * ((k + 1) + 1)) in dom cot & x - (PI * ((k + 1) + 1)) in dom cot & cot . x = cot . (x + (PI * ((k + 1) + 1))) ) )
assume x in dom cot ; :: thesis: ( x + (PI * ((k + 1) + 1)) in dom cot & x - (PI * ((k + 1) + 1)) in dom cot & cot . x = cot . (x + (PI * ((k + 1) + 1))) )
then A9: ( x + (PI * (k + 1)) in dom cot & x - (PI * (k + 1)) in dom cot & cot . x = cot . (x + (PI * (k + 1))) ) by A3, Th1;
then ( x + (PI * (k + 1)) in (dom cos ) /\ ((dom sin ) \ (sin " {0 })) & x - (PI * (k + 1)) in (dom cos ) /\ ((dom sin ) \ (sin " {0 })) ) by RFUNCT_1:def 4;
then ( x + (PI * (k + 1)) in dom cos & x + (PI * (k + 1)) in (dom sin ) \ (sin " {0 }) & x - (PI * (k + 1)) in dom cos & x - (PI * (k + 1)) in (dom sin ) \ (sin " {0 }) ) by XBOOLE_0:def 4;
then ( x + (PI * (k + 1)) in dom cos & x + (PI * (k + 1)) in dom sin & not x + (PI * (k + 1)) in sin " {0 } & x - (PI * (k + 1)) in dom cos & x - (PI * (k + 1)) in dom sin & not x - (PI * (k + 1)) in sin " {0 } ) by XBOOLE_0:def 5;
then ( not sin . (x + (PI * (k + 1))) in {0 } & not sin . (x - (PI * (k + 1))) in {0 } ) by FUNCT_1:def 13;
then B1: ( sin . (x + (PI * (k + 1))) <> 0 & sin . (x - (PI * (k + 1))) <> 0 ) by TARSKI:def 1;
sin . ((x + (PI * (k + 1))) + PI ) = - (sin . (x + (PI * (k + 1)))) by SIN_COS:83;
then not sin . ((x + (PI * (k + 1))) + PI ) in {0 } by TARSKI:def 1, B1;
then ( (x + (PI * (k + 1))) + PI in dom cos & (x + (PI * (k + 1))) + PI in dom sin & not (x + (PI * (k + 1))) + PI in sin " {0 } ) by FUNCT_1:def 13, SIN_COS:27;
then ( (x + (PI * (k + 1))) + PI in dom cos & (x + (PI * (k + 1))) + PI in (dom sin ) \ (sin " {0 }) ) by XBOOLE_0:def 5;
then B2: (x + (PI * (k + 1))) + PI in (dom cos ) /\ ((dom sin ) \ (sin " {0 })) by XBOOLE_0:def 4;
sin . ((x - (PI * ((k + 1) + 1))) + PI ) = - (sin . (x - (PI * ((k + 1) + 1)))) by SIN_COS:83;
then sin . (x - (PI * ((k + 1) + 1))) = - (sin . (x - (PI * (k + 1)))) ;
then not sin . (x - (PI * ((k + 1) + 1))) in {0 } by B1, TARSKI:def 1;
then ( x - (PI * ((k + 1) + 1)) in dom cos & x - (PI * ((k + 1) + 1)) in dom sin & not x - (PI * ((k + 1) + 1)) in sin " {0 } ) by FUNCT_1:def 13, SIN_COS:27;
then ( x - (PI * ((k + 1) + 1)) in dom cos & x - (PI * ((k + 1) + 1)) in (dom sin ) \ (sin " {0 }) ) by XBOOLE_0:def 5;
then B3: x - (PI * ((k + 1) + 1)) in (dom cos ) /\ ((dom sin ) \ (sin " {0 })) by XBOOLE_0:def 4;
then B5: ( x + (PI * ((k + 1) + 1)) in dom cot & x - (PI * ((k + 1) + 1)) in dom cot ) by B2, RFUNCT_1:def 4;
cot . (x + (PI * ((k + 1) + 1))) = (cos . ((x + (PI * (k + 1))) + PI )) / (sin . ((x + (PI * (k + 1))) + PI )) by B5, RFUNCT_1:def 4
.= (- (cos . (x + (PI * (k + 1))))) / (sin . ((x + (PI * (k + 1))) + PI )) by SIN_COS:83
.= (- (cos . (x + (PI * (k + 1))))) / (- (sin . (x + (PI * (k + 1))))) by SIN_COS:83
.= (cos . (x + (PI * (k + 1)))) / (sin . (x + (PI * (k + 1)))) by XCMPLX_1:192
.= cot . x by A9, RFUNCT_1:def 4 ;
hence ( x + (PI * ((k + 1) + 1)) in dom cot & x - (PI * ((k + 1) + 1)) in dom cot & cot . x = cot . (x + (PI * ((k + 1) + 1))) ) by B3, B2, RFUNCT_1:def 4; :: thesis: verum

end;

hence cot is PI * ((k + 1) + 1) -periodic by Th1; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A1, A2);
hence cot is PI * (k + 1) -periodic ; :: thesis: verum