let t be Real; :: thesis: for F being real-valued Function holds
( F is t -periodic iff ( 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) ) ) ) )
let F be real-valued Function; :: thesis: ( F is t -periodic iff ( 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) ) ) ) )
thus ( F is t -periodic implies ( 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) ) ) ) ) :: thesis: ( 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) ) ) implies F is t -periodic )

proof

assume A1: F is t -periodic ; :: thesis: ( 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) ) ) )
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) )

proof

let x be Real; :: thesis: ( x in dom F implies ( x + t in dom F & x - t in dom F & F . x = F . (x + t) ) )
assume x in dom F ; :: thesis: ( x + t in dom F & x - t in dom F & F . x = F . (x + t) )
then (x - t) + t in dom F ;
hence ( x + t in dom F & x - t in dom F & F . x = F . (x + t) ) by A1; :: thesis: verum

end;

hence ( 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 A1; :: thesis: verum

end;

assume 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) ) ) ) ; :: thesis: F is t -periodic
for x being Real holds
( ( x in dom F implies x + t in dom F ) & ( x + t in dom F implies x in dom F ) & ( x in dom F implies F . x = F . (x + t) ) )

proof

let x be Real; :: thesis: ( ( x in dom F implies x + t in dom F ) & ( x + t in dom F implies x in dom F ) & ( x in dom F implies F . x = F . (x + t) ) )
( x in dom F iff x + t in dom F )

proof

( x + t in dom F implies x in dom F )

proof

assume x + t in dom F ; :: thesis: x in dom F
then (x + t) - t in dom F by A2;
hence x in dom F ; :: thesis: verum

end;

hence ( x in dom F iff x + t in dom F ) by A2; :: thesis: verum

end;

hence ( ( x in dom F implies x + t in dom F ) & ( x + t in dom F implies x in dom F ) & ( x in dom F implies F . x = F . (x + t) ) ) by A2; :: thesis: verum

end;

hence F is t -periodic by A2; :: thesis: verum