
theorem
for f be FuzzySet of REAL st
f in {f where f is Function of REAL,REAL :
ex a,b be Real st a<>0 & for th be Real holds f.th= 1/2*sin(a*th+b)+1/2}
holds
f is normalized
proof
 let f be FuzzySet of REAL;
 assume ::A2:
f in {f where f is Function of REAL,REAL :
   ex a,b be Real st a<>0 & for th be Real holds f.th= 1/2*sin(a*th+b)+1/2};
 then consider f2 be Function of REAL,REAL such that
 A3:f=f2 and
 A4: ex a,b be Real st a<>0 & for th be Real holds f2.th= 1/2*sin(a*th+b)+1/2;
 consider a,b be Real such that
 A7: a<>0 and
 A5:for th be Real holds f2.th= 1/2*sin(a*th+b)+1/2 by A4;
 reconsider 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;
   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 thesis by XREAL_0:def 1;
 end;
 hence thesis;
end;
