
theorem
for a,b be Real, f be FuzzySet of 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 a,b be Real, f be FuzzySet of REAL;
 assume A1: a<>0;
 assume A2:( for th be Real holds f.th= 1/2*sin(a*th+b)+1/2);
 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 A2
    .= 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,A1
    .= 1 by SIN_COS:77;
  hence thesis by XREAL_0:def 1;
 end;
 hence thesis;
end;
