let a, b be Real; :: thesis: for k being Nat holds b + (a (#) cos) is (2 * PI) * (k + 1) -periodic
let k be Nat; :: thesis: b + (a (#) cos) is (2 * PI) * (k + 1) -periodic
defpred S₁[ Nat] means b + (a (#) cos) is (2 * PI) * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by Lm27;
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: b + (a (#) cos) is (2 * PI) * (k + 1) -periodic ; :: thesis: S₁[k + 1]
b + (a (#) cos) is (2 * PI) * ((k + 1) + 1) -periodic

proof

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

proof

let x be Real; :: thesis: ( x in dom (b + (a (#) cos)) implies ( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + (a (#) cos)) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + (a (#) cos)) & (b + (a (#) cos)) . x = (b + (a (#) cos)) . (x + ((2 * PI) * ((k + 1) + 1))) ) )
assume x in dom (b + (a (#) cos)) ; :: thesis: ( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + (a (#) cos)) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + (a (#) cos)) & (b + (a (#) cos)) . x = (b + (a (#) cos)) . (x + ((2 * PI) * ((k + 1) + 1))) )
x in REAL by XREAL_0:def 1;
then x in dom (a (#) cos) by SIN_COS:24, VALUED_1:def 5;
then A4: x in dom (b + (a (#) cos)) by VALUED_1:def 2;
( x + ((2 * PI) * ((k + 1) + 1)) in dom cos & x - ((2 * PI) * ((k + 1) + 1)) in dom cos & x + ((2 * PI) * (k + 1)) in dom cos & x - ((2 * PI) * (k + 1)) in dom cos ) by SIN_COS:24, XREAL_0:def 1;
then A5: ( x + ((2 * PI) * ((k + 1) + 1)) in dom (a (#) cos) & x - ((2 * PI) * ((k + 1) + 1)) in dom (a (#) cos) & x + ((2 * PI) * (k + 1)) in dom (a (#) cos) & x - ((2 * PI) * (k + 1)) in dom (a (#) cos) ) by VALUED_1:def 5;
then A6: ( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + (a (#) cos)) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + (a (#) cos)) & x + ((2 * PI) * (k + 1)) in dom (b + (a (#) cos)) & x - ((2 * PI) * (k + 1)) in dom (b + (a (#) cos)) ) by VALUED_1:def 2;
then (b + (a (#) cos)) . (x + ((2 * PI) * ((k + 1) + 1))) = b + ((a (#) cos) . (x + ((2 * PI) * ((k + 1) + 1)))) by VALUED_1:def 2
.= b + (a * (cos . ((x + ((2 * PI) * (k + 1))) + (2 * PI)))) by A5, VALUED_1:def 5
.= b + (a * (cos . (x + ((2 * PI) * (k + 1))))) by SIN_COS:78
.= b + ((a (#) cos) . (x + ((2 * PI) * (k + 1)))) by A5, VALUED_1:def 5
.= (b + (a (#) cos)) . (x + ((2 * PI) * (k + 1))) by A6, VALUED_1:def 2 ;
hence ( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + (a (#) cos)) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + (a (#) cos)) & (b + (a (#) cos)) . x = (b + (a (#) cos)) . (x + ((2 * PI) * ((k + 1) + 1))) ) by A3, A5, A4, VALUED_1:def 2; :: thesis: verum

end;

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