 reserve A for non empty Subset of REAL;

theorem
  for a,c be Real, f be Function of REAL,REAL st
    c > 0 & ( for x be Real holds f.x = max(0, 1 - |. (x-a)/c .|) )
  holds
  f is triangular FuzzySet of REAL
proof
 let a,c be Real;
 let f be Function of REAL,REAL;
 assume B1: c > 0;
 assume A1: for x be Real holds f.x = max(0, 1 - |. (x-a)/c .|);
 deffunc H1(Element of REAL) = In(1 - |. ($1-a)/c .|,REAL);
 consider h being Function of REAL,REAL such that
 A3: for x being Element of REAL holds h.x = H1(x) from FUNCT_2:sch 4;
 A5: for x be Real holds f.x = max(0, min(1, h.x))
 proof
  let x be Real;
  reconsider x as Element of REAL by XREAL_0:def 1;
  A4: h.x = In(1 - |. (x-a)/c .|,REAL) by A3; then
  h.x <= 1 by XREAL_1:43,COMPLEX1:46; then
  max(0, min(1, h.x))
   = max(0, 1 - |. (x-a)/c .|) by A4,XXREAL_0:def 9
  .= f.x by A1;
  hence thesis;
 end;
 reconsider f as FuzzySet of REAL by FUZZY_5:25,A5;
 thus thesis by FUZZY_5:25,A5,FUZZY_5:66,B1,A1;
end;
