let k be Nat; :: thesis: tan is PI * (k + 1) -periodic
defpred S₁[ Nat] means tan is PI * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by L5;
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: tan is PI * (k + 1) -periodic ; :: thesis: S₁[k + 1]
tan is PI * ((k + 1) + 1) -periodic

proof

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

proof

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

end;

hence tan 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 tan is PI * (k + 1) -periodic ; :: thesis: verum