let a, b be Real; :: thesis: for f being Function of REAL,REAL st ( for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) ) holds

f is FuzzySet of REAL

let f be Function of REAL,REAL; :: thesis: ( ( for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) ) implies f is FuzzySet of REAL )

assume for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) ; :: thesis: f is FuzzySet of REAL

then f in { f where f is Function of REAL,REAL, a, b is Real : for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) } ;

hence f is FuzzySet of REAL by Def1, TrF11; :: thesis: verum

f is FuzzySet of REAL

let f be Function of REAL,REAL; :: thesis: ( ( for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) ) implies f is FuzzySet of REAL )

assume for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) ; :: thesis: f is FuzzySet of REAL

then f in { f where f is Function of REAL,REAL, a, b is Real : for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) } ;

hence f is FuzzySet of REAL by Def1, TrF11; :: thesis: verum