
theorem:: GauF02:
for a,b be Real, f be FuzzySet of REAL st
(b<>0 & for x be Real holds f.x= exp(-(x-a)^2/(2*b^2)))
holds
f is strictly-normalized
proof
 let a,b be Real;
 let f be FuzzySet of REAL;
 assume A1: b<>0;
 assume A2: for x be Real holds f.x = exp(-(x-a)^2/(2*b^2));
 ex x being Element of REAL st
 ( f . x = 1 & ( for y being Element of REAL st f . y = 1 holds y = x ) )
 proof
  A4:a is Element of REAL by XREAL_0:def 1;
  take a;
  A3: f.a= exp(-(a-a)^2/(2*b^2)) by A2
    .=1 by SIN_COS3:20;
  for y being Element of REAL st f . y = 1 holds y = a
  proof
   let y be Element of REAL;
   assume f.y=1; then
   exp(-(y-a)^2/(2*b^2)) =1 by A2; then
   (y-a)=0 by A1,EXpReq12;
   hence y=a;
  end;
  hence thesis by A4,A3;
 end;
 hence thesis;
end;
