
theorem
{f where f is Function of REAL,REAL, a,b is Real:
  b > 0 & for x be Real holds f.x = max(0,1-|.(x-a)/b.|)}
    c= Membership_Funcs (REAL)
proof
 {f where f is Function of REAL,REAL, a,b is Real:
   b > 0 & for x be Real holds f.x = max(0,1-|.(x-a)/b.|)}c=
 {TriangularFS (a,b,c) where a,b,c is Real:a < b & b < c}
 proof
  let f be object;
  assume f in {f where f is Function of REAL,REAL, a,b is Real:
      b > 0 & for x be Real holds f.x = max(0,1-|.(x-a)/b.|)}; then
  consider f0 be Function of REAL,REAL, a,b be Real such that
  A1:f=f0 and A2:b>0 and
  A3:for x be Real holds f0.x = max(0,1-|.(x-a)/b.|);
  A4:a-b<a & a<a+b by XREAL_1:44,XREAL_1:29,A2;
  f=TriangularFS (a-b,a,a+b)
  proof
   reconsider f as Function of REAL,REAL by A1;
   A6:dom f = REAL & REAL=dom TriangularFS (a-b,a,a+b) 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
    x in REAL by FUNCT_2:def 1; then
    reconsider x as Real;
    f.x=max(0,1-|.(x-a)/b.|) by A1,A3;
    hence thesis by TR6,A2;
   end;
   hence thesis by FUNCT_1:2, A6;
  end;
  hence thesis by A4;
 end;
 hence thesis by TR2XX;
end;
