let g be object ; :: according to TARSKI:def 3 :: thesis: ( not g in { f where f is Function of REAL,REAL, a, b is Real : for x being Real holds f . x = max (0,(min (1,((AffineMap (a,b)) . x)))) } or g in Membership_Funcs REAL )

assume g in { f where f is Function of REAL,REAL, a, b is Real : for x being Real holds f . x = max (0,(min (1,((AffineMap (a,b)) . x)))) } ; :: thesis: g in Membership_Funcs REAL

then consider f being Function of REAL,REAL, a, b being Real such that

A1: f = g and

A2: for x being Real holds f . x = max (0,(min (1,((AffineMap (a,b)) . x)))) ;

g is FuzzySet of REAL by A1, A2, MM40;

hence g in Membership_Funcs REAL by Def1; :: thesis: verum

assume g in { f where f is Function of REAL,REAL, a, b is Real : for x being Real holds f . x = max (0,(min (1,((AffineMap (a,b)) . x)))) } ; :: thesis: g in Membership_Funcs REAL

then consider f being Function of REAL,REAL, a, b being Real such that

A1: f = g and

A2: for x being Real holds f . x = max (0,(min (1,((AffineMap (a,b)) . x)))) ;

g is FuzzySet of REAL by A1, A2, MM40;

hence g in Membership_Funcs REAL by Def1; :: thesis: verum