
theorem MM2:
{f where f is Function of REAL,REAL :
ex a,b be Real st for x be Real holds f.x= max(0, cos(a*x+b))}
 c= Membership_Funcs (REAL)
proof
 let x be object;
 assume x in {f where f is Function of REAL,REAL :
 ex a,b be Real st for th be Real holds f.th= max(0, cos(a*th+b))}; then
 consider f be Function of REAL,REAL such that
 A1: x=f and
 A2: ex a,b be Real st for th be Real holds f.th= max(0, cos(a*th+b));
 rng f c= [.0,1.]
 proof
  let y be object;
  assume y in rng f;then
  consider th be object such that
  B2: th in REAL and B3: y = f . th by FUNCT_2:11;
  reconsider th as Real by B2;
  consider a,b be Real such that
  B1:for th0 be Real holds f.th0= max(0, cos(a*th0+b)) by A2;
  cos(a*th+b) <= 1 by SIN_COS6:6;
  then
  B4: 0 <= max(0, cos(a*th+b)) &
  max(0, cos(a*th+b)) <= 1 by XXREAL_0:25,XXREAL_0:28;
  y = max(0, cos(a*th+b)) by B1,B3;
  hence thesis by B4;
 end;
 then f is [.0,1.] -valued;
 hence thesis by Def1,A1;
end;
