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 Lm11;
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 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; :: 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 A4: ( 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 1;
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 7;
then A5: ( 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:78;
then not sin . ((x + (PI * (k + 1))) + PI) in {0} by A5, TARSKI:def 1;
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 7, SIN_COS:24, XREAL_0:def 1;
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 A6: (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:78;
then sin . (x - (PI * ((k + 1) + 1))) = - (sin . (x - (PI * (k + 1)))) ;
then not sin . (x - (PI * ((k + 1) + 1))) in {0} by A5, 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 7, SIN_COS:24, XREAL_0:def 1;
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 A7: x - (PI * ((k + 1) + 1)) in (dom cos) /\ ((dom sin) \ (sin " {0})) by XBOOLE_0:def 4;
then ( x + (PI * ((k + 1) + 1)) in dom cot & x - (PI * ((k + 1) + 1)) in dom cot ) by A6, RFUNCT_1:def 1;
then cot . (x + (PI * ((k + 1) + 1))) = (cos . ((x + (PI * (k + 1))) + PI)) / (sin . ((x + (PI * (k + 1))) + PI)) by RFUNCT_1:def 1
.= (- (cos . (x + (PI * (k + 1))))) / (sin . ((x + (PI * (k + 1))) + PI)) by SIN_COS:78
.= (- (cos . (x + (PI * (k + 1))))) / (- (sin . (x + (PI * (k + 1))))) by SIN_COS:78
.= (cos . (x + (PI * (k + 1)))) / (sin . (x + (PI * (k + 1)))) by XCMPLX_1:191
.= cot . x by A4, RFUNCT_1:def 1 ;
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 A7, A6, RFUNCT_1:def 1; :: 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