let k be Nat; :: thesis: cos ^2 is PI * (k + 1) -periodic
defpred S₁[ Nat] means cos ^2 is PI * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by Lm21;
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 ^2 is PI * (k + 1) -periodic ; :: thesis: S₁[k + 1]
cos ^2 is PI * ((k + 1) + 1) -periodic

proof

for x being Real st x in dom (cos ^2) holds
( x + (PI * ((k + 1) + 1)) in dom (cos ^2) & x - (PI * ((k + 1) + 1)) in dom (cos ^2) & (cos ^2) . x = (cos ^2) . (x + (PI * ((k + 1) + 1))) )

proof

let x be Real; :: thesis: ( x in dom (cos ^2) implies ( x + (PI * ((k + 1) + 1)) in dom (cos ^2) & x - (PI * ((k + 1) + 1)) in dom (cos ^2) & (cos ^2) . x = (cos ^2) . (x + (PI * ((k + 1) + 1))) ) )
assume A4: x in dom (cos ^2) ; :: thesis: ( x + (PI * ((k + 1) + 1)) in dom (cos ^2) & x - (PI * ((k + 1) + 1)) in dom (cos ^2) & (cos ^2) . x = (cos ^2) . (x + (PI * ((k + 1) + 1))) )
A5: ( x + (PI * ((k + 1) + 1)) in dom cos & x - (PI * ((k + 1) + 1)) in dom cos ) by SIN_COS:24, XREAL_0:def 1;
(cos ^2) . (x + (PI * ((k + 1) + 1))) = (cos . ((x + (PI * (k + 1))) + PI)) ^2 by VALUED_1:11
.= (- (cos . (x + (PI * (k + 1))))) ^2 by SIN_COS:78
.= (cos . (x + (PI * (k + 1)))) ^2
.= (cos ^2) . (x + (PI * (k + 1))) by VALUED_1:11 ;
hence ( x + (PI * ((k + 1) + 1)) in dom (cos ^2) & x - (PI * ((k + 1) + 1)) in dom (cos ^2) & (cos ^2) . x = (cos ^2) . (x + (PI * ((k + 1) + 1))) ) by A3, A4, A5, VALUED_1:11; :: thesis: verum

end;

hence cos ^2 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 ^2 is PI * (k + 1) -periodic ; :: thesis: verum