let F be Function of REAL,REAL; :: thesis:
let a, b, c, d be Real; :: thesis:
let i be Integer; :: thesis:
assume A0: ( a <> 0 & i <> 0 ) ; :: thesis:
assume A2: for x being Real holds F . x = max (0,(min (1,((c * (sin ((a * x) + b))) + d)))) ; :: thesis:
for x being Real st x in dom F holds
( x + (((2 * PI) / a) * i) in dom F & x - (((2 * PI) / a) * i) in dom F & F . x = F . (x + (((2 * PI) / a) * i)) )

let x be Real; :: thesis:
assume x in dom F ; :: thesis:
A3A: ( x + (((2 * PI) / a) * i) in REAL & x - (((2 * PI) / a) * i) in REAL ) by XREAL_0:def 1;
S1: sin is (2 * PI) * i -periodic by FUNCT_9:21, A0;
SS: (a * x) + b in dom sin by SIN_COS:24, XREAL_0:def 1;
F . (x + (((2 * PI) / a) * i)) = max (0,(min (1,((c * (sin ((a * (x + (((2 * PI) / a) * i))) + b))) + d)))) by A2
.= max (0,(min (1,((c * (sin (((a * x) + ((a * ((2 * PI) / a)) * i)) + b))) + d))))
.= max (0,(min (1,((c * (sin (((a * x) + (((a / a) * (2 * PI)) * i)) + b))) + d)))) by XCMPLX_1:75
.= max (0,(min (1,((c * (sin (((a * x) + ((1 * (2 * PI)) * i)) + b))) + d)))) by XCMPLX_1:60, A0
.= max (0,(min (1,((c * (sin . (((a * x) + b) + ((2 * PI) * i)))) + d)))) by SIN_COS:def 17
.= max (0,(min (1,((c * (sin . ((a * x) + b))) + d)))) by FUNCT_9:1, SS, S1
.= max (0,(min (1,((c * (sin ((a * x) + b))) + d)))) by SIN_COS:def 17 ;
hence ( x + (((2 * PI) / a) * i) in dom F & x - (((2 * PI) / a) * i) in dom F & F . x = F . (x + (((2 * PI) / a) * i)) ) by FUNCT_2:def 1, A3A, A2; :: thesis:

end;