let t1, t2 be Real; :: thesis: for F being real-valued Function st t1 - t2 <> 0 & ( for x being Real st x in dom F holds
( x + t1 in dom F & x - t1 in dom F & x + t2 in dom F & x - t2 in dom F & F . (x + t1) = F . (x + t2) ) ) holds
( F is t1 - t2 -periodic & F is periodic )
let F be real-valued Function; :: thesis: ( t1 - t2 <> 0 & ( for x being Real st x in dom F holds
( x + t1 in dom F & x - t1 in dom F & x + t2 in dom F & x - t2 in dom F & F . (x + t1) = F . (x + t2) ) ) implies ( F is t1 - t2 -periodic & F is periodic ) )
assume that
A1: t1 - t2 <> 0 and
A2: for x being Real st x in dom F holds
( x + t1 in dom F & x - t1 in dom F & x + t2 in dom F & x - t2 in dom F & F . (x + t1) = F . (x + t2) ) ; :: thesis: ( F is t1 - t2 -periodic & F is periodic )
for x being Real st x in dom F holds
( x + (t1 - t2) in dom F & x - (t1 - t2) in dom F & F . x = F . (x + (t1 - t2)) )

proof

let x be Real; :: thesis: ( x in dom F implies ( x + (t1 - t2) in dom F & x - (t1 - t2) in dom F & F . x = F . (x + (t1 - t2)) ) )
assume x in dom F ; :: thesis: ( x + (t1 - t2) in dom F & x - (t1 - t2) in dom F & F . x = F . (x + (t1 - t2)) )
then A3: ( x + t1 in dom F & x - t1 in dom F & x + t2 in dom F & x - t2 in dom F ) by A2;
then A4: ( (x + t1) - t2 in dom F & (x - t1) + t2 in dom F ) by A2;
F . (x + (t1 - t2)) = F . ((x - t2) + t1)
.= F . ((x - t2) + t2) by A2, A3
.= F . x ;
hence ( x + (t1 - t2) in dom F & x - (t1 - t2) in dom F & F . x = F . (x + (t1 - t2)) ) by A4; :: thesis: verum

end;

then F is t1 - t2 -periodic by Th1, A1;
hence ( F is t1 - t2 -periodic & F is periodic ) ; :: thesis: verum