let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { f where f is Function of REAL,REAL : ex a, b being Real st
for th being Real holds f . th = ((1 / 2) * (cos ((a * th) + b))) + (1 / 2) } or x in Membership_Funcs REAL )
assume x in { f where f is Function of REAL,REAL : ex a, b being Real st
for th being Real holds f . th = ((1 / 2) * (cos ((a * th) + b))) + (1 / 2) } ; :: thesis: x in Membership_Funcs REAL
then consider f being Function of REAL,REAL such that
A1: x = f and
A2: ex a, b being Real st
for th being Real holds f . th = ((1 / 2) * (cos ((a * th) + b))) + (1 / 2) ;
rng f c= [.0,1.]

proof

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 th being object such that
B2: th in REAL and
B3: y = f . th by FUNCT_2:11;
reconsider th = th as Real by B2;
consider a, b being Real such that
B1: for th0 being Real holds f . th0 = ((1 / 2) * (cos ((a * th0) + b))) + (1 / 2) by A2;
|.(cos ((a * th) + b)).| <= 1 by SIN_COS:27;
then ( - 1 <= cos ((a * th) + b) & cos ((a * th) + b) <= 1 ) by ABSVALUE:5;
then ( (1 / 2) * (- 1) <= (1 / 2) * (cos ((a * th) + b)) & (1 / 2) * (cos ((a * th) + b)) <= (1 / 2) * 1 ) by XREAL_1:64;
then B4: ( (- (1 / 2)) + (1 / 2) <= (((cos ((a * th) + b)) * 1) / 2) + (1 / 2) & ((1 / 2) * (cos ((a * th) + b))) + (1 / 2) <= (1 / 2) + (1 / 2) ) by XREAL_1:7;
y = ((1 / 2) * (cos ((a * th) + b))) + (1 / 2) by B1, B3;
hence y in [.0,1.] by B4; :: thesis: verum

end;

then f is [.0,1.] -valued ;
hence x in Membership_Funcs REAL by Def1, A1; :: thesis: verum