let k be Nat; :: thesis: |.cos.| is PI * (k + 1) -periodic
defpred S₁[ Nat] means |.cos.| is PI * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by L8;
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: |.cos.| is PI * (k + 1) -periodic ; :: thesis: S₁[k + 1]
|.cos.| is PI * ((k + 1) + 1) -periodic

proof

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

proof

let x be real number ; :: thesis: ( x in dom |.cos.| implies ( x + (PI * ((k + 1) + 1)) in dom |.cos.| & x - (PI * ((k + 1) + 1)) in dom |.cos.| & |.cos.| . x = |.cos.| . (x + (PI * ((k + 1) + 1))) ) )
assume A7: x in dom |.cos.| ; :: thesis: ( x + (PI * ((k + 1) + 1)) in dom |.cos.| & x - (PI * ((k + 1) + 1)) in dom |.cos.| & |.cos.| . x = |.cos.| . (x + (PI * ((k + 1) + 1))) )
then A8: ( x + (PI * (k + 1)) in dom |.cos.| & x - (PI * (k + 1)) in dom |.cos.| ) by A3, Th1;
A9: ( x + (PI * ((k + 1) + 1)) in dom cos & x - (PI * ((k + 1) + 1)) in dom cos ) by SIN_COS:27;
then ( x + (PI * ((k + 1) + 1)) in dom |.cos.| & x - (PI * ((k + 1) + 1)) in dom |.cos.| ) by VALUED_1:def 11;
then |.cos.| . (x + (PI * ((k + 1) + 1))) = |.(cos . ((x + (PI * (k + 1))) + PI)).| by VALUED_1:def 11
.= |.(- (cos . (x + (PI * (k + 1))))).| by SIN_COS:83
.= |.(cos . (x + (PI * (k + 1)))).| by COMPLEX1:138
.= |.cos.| . (x + (PI * (k + 1))) by A8, VALUED_1:def 11 ;
hence ( x + (PI * ((k + 1) + 1)) in dom |.cos.| & x - (PI * ((k + 1) + 1)) in dom |.cos.| & |.cos.| . x = |.cos.| . (x + (PI * ((k + 1) + 1))) ) by A3, Th1, A7, A9, VALUED_1:def 11; :: thesis: verum

end;

hence |.cos.| 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 |.cos.| is PI * (k + 1) -periodic ; :: thesis: verum