let t, r be Real; :: thesis: for F being real-valued Function st F is t -periodic holds

r + F is t -periodic

let F be real-valued Function; :: thesis: ( F is t -periodic implies r + F is t -periodic )

assume A1: F is t -periodic ; :: thesis: r + F is t -periodic

then A2: ( t <> 0 & ( for x being Real st x in dom F holds

( x + t in dom F & x - t in dom F & F . x = F . (x + t) ) ) ) by Th1;

for x being Real st x in dom (r + F) holds

( x + t in dom (r + F) & x - t in dom (r + F) & (r + F) . x = (r + F) . (x + t) )

r + F is t -periodic

let F be real-valued Function; :: thesis: ( F is t -periodic implies r + F is t -periodic )

assume A1: F is t -periodic ; :: thesis: r + F is t -periodic

then A2: ( t <> 0 & ( for x being Real st x in dom F holds

( x + t in dom F & x - t in dom F & F . x = F . (x + t) ) ) ) by Th1;

for x being Real st x in dom (r + F) holds

( x + t in dom (r + F) & x - t in dom (r + F) & (r + F) . x = (r + F) . (x + t) )

proof

hence
r + F is t -periodic
by A2, Th1; :: thesis: verum
let x be Real; :: thesis: ( x in dom (r + F) implies ( x + t in dom (r + F) & x - t in dom (r + F) & (r + F) . x = (r + F) . (x + t) ) )

assume A3: x in dom (r + F) ; :: thesis: ( x + t in dom (r + F) & x - t in dom (r + F) & (r + F) . x = (r + F) . (x + t) )

then A4: x in dom F by VALUED_1:def 2;

then A5: ( x + t in dom F & x - t in dom F ) by A1, Th1;

then A6: ( x + t in dom (r + F) & x - t in dom (r + F) ) by VALUED_1:def 2;

(r + F) . x = r + (F . x) by A3, VALUED_1:def 2

.= r + (F . (x + t)) by A1, A4

.= (r + F) . (x + t) by A6, VALUED_1:def 2 ;

hence ( x + t in dom (r + F) & x - t in dom (r + F) & (r + F) . x = (r + F) . (x + t) ) by A5, VALUED_1:def 2; :: thesis: verum

end;assume A3: x in dom (r + F) ; :: thesis: ( x + t in dom (r + F) & x - t in dom (r + F) & (r + F) . x = (r + F) . (x + t) )

then A4: x in dom F by VALUED_1:def 2;

then A5: ( x + t in dom F & x - t in dom F ) by A1, Th1;

then A6: ( x + t in dom (r + F) & x - t in dom (r + F) ) by VALUED_1:def 2;

(r + F) . x = r + (F . x) by A3, VALUED_1:def 2

.= r + (F . (x + t)) by A1, A4

.= (r + F) . (x + t) by A6, VALUED_1:def 2 ;

hence ( x + t in dom (r + F) & x - t in dom (r + F) & (r + F) . x = (r + F) . (x + t) ) by A5, VALUED_1:def 2; :: thesis: verum