
theorem TrF22:
for f be FuzzySet of REAL
for a,b be Real st a<>0 & for th be Real holds f.th= 1/2*cos(a*th+b)+1/2
holds
f is normalized
proof
 let f be FuzzySet of REAL;
 let a,b be Real;
 assume A1:a<>0;
 assume A2:for th be Real holds f.th= 1/2*cos(a*th+b)+1/2;
 ex x being Element of REAL st f . x = 1
 proof
  take (-b)/a;
  f.((-b)/a)=1/2*cos(a*((-b)/a)+b)+1/2 by A2
   .= 1/2*cos(a/a*(-b)+b)+1/2 by XCMPLX_1:75
   .= 1/2*cos(1*(-b)+b)+1/2 by XCMPLX_1:60,A1
   .= 1 by SIN_COS:31;
  hence thesis by XREAL_0:def 1;
 end;
 hence thesis;
end;
