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 Lm10;
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 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; :: 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 A4: ( 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 1;
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 7;
then A5: ( 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:78;
then not cos . ((x + (PI * (k + 1))) + PI) in {0} by A5, TARSKI:def 1;
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 7, SIN_COS:24, XREAL_0:def 1;
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 A6: (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:78;
then cos . (x - (PI * ((k + 1) + 1))) = - (cos . (x - (PI * (k + 1)))) ;
then not cos . (x - (PI * ((k + 1) + 1))) in {0} by A5, 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 7, SIN_COS:24, XREAL_0:def 1;
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 A7: 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 A6, RFUNCT_1:def 1;
then tan . (x + (PI * ((k + 1) + 1))) = (sin . ((x + (PI * (k + 1))) + PI)) / (cos . ((x + (PI * (k + 1))) + PI)) by RFUNCT_1:def 1
.= (- (sin . (x + (PI * (k + 1))))) / (cos . ((x + (PI * (k + 1))) + PI)) by SIN_COS:78
.= (- (sin . (x + (PI * (k + 1))))) / (- (cos . (x + (PI * (k + 1))))) by SIN_COS:78
.= (sin . (x + (PI * (k + 1)))) / (cos . (x + (PI * (k + 1)))) by XCMPLX_1:191
.= tan . x by A4, RFUNCT_1:def 1 ;
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 A7, A6, RFUNCT_1:def 1; :: 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