let a, b be Real; :: thesis: for k being Nat holds b + () is (2 * PI) * (k + 1) -periodic
let k be Nat; :: thesis: b + () is (2 * PI) * (k + 1) -periodic
defpred S1[ Nat] means b + () is (2 * PI) * (\$1 + 1) -periodic ;
A1: S1[ 0 ] by Lm26;
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 + () is (2 * PI) * (k + 1) -periodic ; :: thesis: S1[k + 1]
b + () is (2 * PI) * ((k + 1) + 1) -periodic
proof
for x being Real st x in dom (b + ()) holds
( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & (b + ()) . x = (b + ()) . (x + ((2 * PI) * ((k + 1) + 1))) )
proof
let x be Real; :: thesis: ( x in dom (b + ()) implies ( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & (b + ()) . x = (b + ()) . (x + ((2 * PI) * ((k + 1) + 1))) ) )
assume x in dom (b + ()) ; :: thesis: ( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & (b + ()) . x = (b + ()) . (x + ((2 * PI) * ((k + 1) + 1))) )
x in REAL by XREAL_0:def 1;
then x in dom () by ;
then A4: x in dom (b + ()) 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 ;
then A5: ( x + ((2 * PI) * ((k + 1) + 1)) in dom () & x - ((2 * PI) * ((k + 1) + 1)) in dom () & x + ((2 * PI) * (k + 1)) in dom () & x - ((2 * PI) * (k + 1)) in dom () ) by VALUED_1:def 5;
then A6: ( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & x + ((2 * PI) * (k + 1)) in dom (b + ()) & x - ((2 * PI) * (k + 1)) in dom (b + ()) ) by VALUED_1:def 2;
then (b + ()) . (x + ((2 * PI) * ((k + 1) + 1))) = b + (() . (x + ((2 * PI) * ((k + 1) + 1)))) by VALUED_1:def 2
.= b + (a * (sin . ((x + ((2 * PI) * (k + 1))) + (2 * PI)))) by
.= b + (a * (sin . (x + ((2 * PI) * (k + 1))))) by SIN_COS:78
.= b + (() . (x + ((2 * PI) * (k + 1)))) by
.= (b + ()) . (x + ((2 * PI) * (k + 1))) by ;
hence ( x + ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & x - ((2 * PI) * ((k + 1) + 1)) in dom (b + ()) & (b + ()) . x = (b + ()) . (x + ((2 * PI) * ((k + 1) + 1))) ) by ; :: thesis: verum
end;
hence b + () 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 + () is (2 * PI) * (k + 1) -periodic ; :: thesis: verum