
theorem:: TR8:
for a,b be Real, f be FuzzySet of REAL st
b > 0 & (for x be Real holds f.x = max(0,1-|.(x-a)/b.|))
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 = max(0,1-|.(x-a)/b.|);
 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 = max(0,1-|.(a-a)/b.|) by A2
         .= max(0,1-0) by ABSVALUE:def 1
         .= 1 by XXREAL_0:def 10;
  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
   max(0,1-|.(y-a)/b.|)=1 by A2; then
   1-|.(y-a)/b.|=1 by XXREAL_0:16; then
   y-a=0 by A1,ABSVALUE:2;
   hence y=a;
  end;
  hence thesis by A4,A3;
 end;
 hence thesis;
end;
