
theorem
for F being Function of REAL,REAL, a,b,c,d being Real st
for x be Real holds F.x= max(0,min(1, c*sin(a*x+b)+d))
holds
F is periodic
proof
 let F be Function of REAL,REAL;
 let a,b,c,d be Real;
 assume A0: for x be Real holds F.x= max(0,min(1, c*sin(a*x+b)+d));
 ex t being Real st F is t -periodic
 proof
 per cases;
 suppose A1:a<>0;
  take (2 * PI)/a * 1;
  thus thesis by TrF160,A0,A1;
 end;
 suppose A5:a=0;
  take 1;
  for x being Real st x in dom F holds
  ( x + 1 in dom F & x - 1 in dom F & F . x = F . (x + 1) )
  proof
   let x be Real;
   assume x in dom F;
 A4A:  x + 1 in REAL & x - 1 in REAL by XREAL_0:def 1;
   F . x = max(0,min(1, c*sin(a*x+b)+d)) by A0
   .= max(0,min(1, c*sin(0*(x+1)+b)+d)) by A5;
   hence thesis by A4A,FUNCT_2:def 1,A0,A5;
  end;
  hence F is 1 -periodic by FUNCT_9:1;
 end;
 end;
 hence thesis by FUNCT_9:def 2;
end;
