let f be Function of REAL,REAL; :: thesis: for a, b, c, d being Real st ( for x being Real holds f . x = max (0,(min (1,((c * (arctan ((a * x) + b))) + d)))) ) holds

f is FuzzySet of REAL

let a, b, c, d be Real; :: thesis: ( ( for x being Real holds f . x = max (0,(min (1,((c * (arctan ((a * x) + b))) + d)))) ) implies f is FuzzySet of REAL )

assume for x being Real holds f . x = max (0,(min (1,((c * (arctan ((a * x) + b))) + d)))) ; :: thesis: f is FuzzySet of REAL

then f in { f where f is Function of REAL,REAL, a, b, c, d is Real : for x being Real holds f . x = max (0,(min (1,((c * (arctan ((a * x) + b))) + d)))) } ;

then f in Membership_Funcs REAL by MM60, MM41;

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

f is FuzzySet of REAL

let a, b, c, d be Real; :: thesis: ( ( for x being Real holds f . x = max (0,(min (1,((c * (arctan ((a * x) + b))) + d)))) ) implies f is FuzzySet of REAL )

assume for x being Real holds f . x = max (0,(min (1,((c * (arctan ((a * x) + b))) + d)))) ; :: thesis: f is FuzzySet of REAL

then f in { f where f is Function of REAL,REAL, a, b, c, d is Real : for x being Real holds f . x = max (0,(min (1,((c * (arctan ((a * x) + b))) + d)))) } ;

then f in Membership_Funcs REAL by MM60, MM41;

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