
theorem
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 triangular
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.|);
 take a-b,a,a+b;
 A4: dom TriangularFS (a-b,a,a+b) = REAL &
 dom f = REAL by FUNCT_2:def 1;
 for x being object st x in dom f holds
 f . x = TriangularFS (a-b,a,a+b).x
 proof
  let x be object;
  assume x in dom f; then
  reconsider x as Real by A4;
  f . x = max(0,1-|.(x-a)/b.|) by A2
       .= TriangularFS (a-b,a,a+b).x by A1,TR6;
  hence thesis;
 end;
 hence thesis by FUNCT_1:2,A4;
end;
