
theorem MM5a:
{f where f is Function of REAL,REAL, a,b,c,d is Real:
 for x be Real holds f.x= max(0,min(1, c*sin(a*x+b)+d))}
 c= Membership_Funcs (REAL)
proof
 let g be object;
 assume g in {f where f is Function of REAL,REAL, a,b,c,d is Real:
 for x be Real holds f.x= max(0,min(1, c*sin(a*x+b)+d))};
 then
 consider f be Function of REAL,REAL, a,b,c,d be Real such that
 A1:f=g and
 A2:for x be Real holds f.x= max(0,min(1, c*sin(a*x+b)+d));
 f is FuzzySet of REAL
 proof
  rng f c= [.0,1.]
  proof
   let y be object;
   assume y in rng f; then
   consider x be object such that
   B2: x in REAL and B3: y = f.x by FUNCT_2:11;
   reconsider x as Real by B2;
   B4:y = max(0,min(1, c*sin(a*x+b)+d)) by A2,B3;
   min(1, c*sin(a*x+b)+d)<=1 by XXREAL_0:17; then
   0<=max(0,min(1, c*sin(a*x+b)+d))
   & max(0,min(1, c*sin(a*x+b)+d))<=1 by XXREAL_0:28,XXREAL_0:25;
   hence thesis by B4;
  end;
  then
  f is [.0,1.] -valued;
  hence thesis;
 end;
 hence thesis by Def1,A1;
end;
