let f be FuzzySet of REAL; :: thesis: ( f in { f where f is Function of REAL,REAL : ex a, b being Real st
( a <> 0 & ( for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) ) ) } implies f is normalized )
assume f in { f where f is Function of REAL,REAL : ex a, b being Real st
( a <> 0 & ( for th being Real holds f . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) ) ) } ; :: thesis: f is normalized
then consider f2 being Function of REAL,REAL such that
A3: f = f2 and
A4: ex a, b being Real st
( a <> 0 & ( for th being Real holds f2 . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) ) ) ;
consider a, b being Real such that
A7: a <> 0 and
A5: for th being Real holds f2 . th = ((1 / 2) * (sin ((a * th) + b))) + (1 / 2) by A4;
reconsider a = a as Element of REAL by XREAL_0:def 1;
ex x being Element of REAL st f . x = 1

proof

take ((PI / 2) - b) / a ; :: thesis: ( ((PI / 2) - b) / a is Element of REAL & f . (((PI / 2) - b) / a) = 1 )
f . (((PI / 2) - b) / a) = ((1 / 2) * (sin ((a * (((PI / 2) - b) / a)) + b))) + (1 / 2) by A5, A3
.= ((1 / 2) * (sin (((a / a) * ((PI / 2) - b)) + b))) + (1 / 2) by XCMPLX_1:75
.= ((1 / 2) * (sin ((1 * ((PI / 2) - b)) + b))) + (1 / 2) by XCMPLX_1:60, A7
.= 1 by SIN_COS:77 ;
hence ( ((PI / 2) - b) / a is Element of REAL & f . (((PI / 2) - b) / a) = 1 ) by XREAL_0:def 1; :: thesis: verum

end;

hence f is normalized ; :: thesis: verum