let g be Function of REAL,REAL; :: thesis: { f where f is Function of REAL,REAL : for x being Real holds f . x = min (1,(max (0,(g . x)))) } c= Membership_Funcs REAL

let f0 be object ; :: according to TARSKI:def 3 :: thesis: ( not f0 in { f where f is Function of REAL,REAL : for x being Real holds f . x = min (1,(max (0,(g . x)))) } or f0 in Membership_Funcs REAL )

assume f0 in { f where f is Function of REAL,REAL : for x being Real holds f . x = min (1,(max (0,(g . x)))) } ; :: thesis: f0 in Membership_Funcs REAL

then consider f being Function of REAL,REAL such that

A1: f0 = f and

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

rng f c= [.0,1.]

hence f0 in Membership_Funcs REAL by Def1, A1; :: thesis: verum

let f0 be object ; :: according to TARSKI:def 3 :: thesis: ( not f0 in { f where f is Function of REAL,REAL : for x being Real holds f . x = min (1,(max (0,(g . x)))) } or f0 in Membership_Funcs REAL )

assume f0 in { f where f is Function of REAL,REAL : for x being Real holds f . x = min (1,(max (0,(g . x)))) } ; :: thesis: f0 in Membership_Funcs REAL

then consider f being Function of REAL,REAL such that

A1: f0 = f and

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

rng f c= [.0,1.]

proof

then
f is [.0,1.] -valued
;
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng f or y in [.0,1.] )

assume y in rng f ; :: thesis: y in [.0,1.]

then consider x being object such that

B2: x in REAL and

B3: y = f . x by FUNCT_2:11;

reconsider x = x as Real by B2;

B4: y = min (1,(max (0,(g . x)))) by A2, B3;

0 <= max (0,(g . x)) by XXREAL_0:25;

then ( 0 <= min (1,(max (0,(g . x)))) & min (1,(max (0,(g . x)))) <= 1 ) by XXREAL_0:20, XXREAL_0:17;

hence y in [.0,1.] by B4; :: thesis: verum

end;assume y in rng f ; :: thesis: y in [.0,1.]

then consider x being object such that

B2: x in REAL and

B3: y = f . x by FUNCT_2:11;

reconsider x = x as Real by B2;

B4: y = min (1,(max (0,(g . x)))) by A2, B3;

0 <= max (0,(g . x)) by XXREAL_0:25;

then ( 0 <= min (1,(max (0,(g . x)))) & min (1,(max (0,(g . x)))) <= 1 ) by XXREAL_0:20, XXREAL_0:17;

hence y in [.0,1.] by B4; :: thesis: verum

hence f0 in Membership_Funcs REAL by Def1, A1; :: thesis: verum