let b, a be real number ; :: thesis: for k being Nat holds b + (a (#) sin ) is (2 * PI ) * (k + 1) -periodic
let k be Nat; :: thesis: b + (a (#) sin ) is (2 * PI ) * (k + 1) -periodic
defpred S₁[ Nat] means b + (a (#) sin ) is (2 * PI ) * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by L14;
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 (#) sin ) is (2 * PI ) * (k + 1) -periodic ; :: thesis: S₁[k + 1]
b + (a (#) sin ) is (2 * PI ) * ((k + 1) + 1) -periodic

proof

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

proof

let x be real number ; :: thesis: ( x in dom (b + (a (#) sin )) implies ( x + ((2 * PI ) * ((k + 1) + 1)) in dom (b + (a (#) sin )) & x - ((2 * PI ) * ((k + 1) + 1)) in dom (b + (a (#) sin )) & (b + (a (#) sin )) . x = (b + (a (#) sin )) . (x + ((2 * PI ) * ((k + 1) + 1))) ) )
assume x in dom (b + (a (#) sin )) ; :: thesis: ( x + ((2 * PI ) * ((k + 1) + 1)) in dom (b + (a (#) sin )) & x - ((2 * PI ) * ((k + 1) + 1)) in dom (b + (a (#) sin )) & (b + (a (#) sin )) . x = (b + (a (#) sin )) . (x + ((2 * PI ) * ((k + 1) + 1))) )
x in REAL by XREAL_0:def 1;
then x in dom (a (#) sin ) by SIN_COS:27, VALUED_1:def 5;
then A9: x in dom (b + (a (#) sin )) by VALUED_1:def 2;
( x + ((2 * PI ) * ((k + 1) + 1)) in dom sin & x - ((2 * PI ) * ((k + 1) + 1)) in dom sin & x + ((2 * PI ) * (k + 1)) in dom sin & x - ((2 * PI ) * (k + 1)) in dom sin ) by SIN_COS:27;
then A11: ( x + ((2 * PI ) * ((k + 1) + 1)) in dom (a (#) sin ) & x - ((2 * PI ) * ((k + 1) + 1)) in dom (a (#) sin ) & x + ((2 * PI ) * (k + 1)) in dom (a (#) sin ) & x - ((2 * PI ) * (k + 1)) in dom (a (#) sin ) ) by VALUED_1:def 5;
then A12: ( x + ((2 * PI ) * ((k + 1) + 1)) in dom (b + (a (#) sin )) & x - ((2 * PI ) * ((k + 1) + 1)) in dom (b + (a (#) sin )) & x + ((2 * PI ) * (k + 1)) in dom (b + (a (#) sin )) & x - ((2 * PI ) * (k + 1)) in dom (b + (a (#) sin )) ) by VALUED_1:def 2;
then (b + (a (#) sin )) . (x + ((2 * PI ) * ((k + 1) + 1))) = b + ((a (#) sin ) . (x + ((2 * PI ) * ((k + 1) + 1)))) by VALUED_1:def 2
.= b + (a * (sin . ((x + ((2 * PI ) * (k + 1))) + (2 * PI )))) by VALUED_1:def 5, A11
.= b + (a * (sin . (x + ((2 * PI ) * (k + 1))))) by SIN_COS:83
.= b + ((a (#) sin ) . (x + ((2 * PI ) * (k + 1)))) by VALUED_1:def 5, A11
.= (b + (a (#) sin )) . (x + ((2 * PI ) * (k + 1))) by VALUED_1:def 2, A12 ;
hence ( x + ((2 * PI ) * ((k + 1) + 1)) in dom (b + (a (#) sin )) & x - ((2 * PI ) * ((k + 1) + 1)) in dom (b + (a (#) sin )) & (b + (a (#) sin )) . x = (b + (a (#) sin )) . (x + ((2 * PI ) * ((k + 1) + 1))) ) by A3, Th1, A11, A9, VALUED_1:def 2; :: thesis: verum

end;

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