
theorem
for a,b be Real, f be Function of REAL,REAL st
(for x be Real holds f.x = max(0,1-|.(x-a)/b.|))
holds f is FuzzySet of REAL
proof
 let a,b be Real, f be Function of REAL,REAL;
 assume B2: for x be Real holds
 f.x = max(0,1-|.(x-a)/b.|);
 rng f c= [.0,1.]
 proof
  let y be object;
  assume y in rng f; then
  consider x be object such that
  B1: x in REAL and B3: y = f . x by FUNCT_2:11;
  reconsider x as Real by B1;
  0 <= max(0,1-|.(x-a)/b.|) & max(0,1-|.(x-a)/b.|) <= 1
  proof
   0 <=|.(x-a)/b.| by COMPLEX1:46; then
   1+0 <= 1+|.(x-a)/b.| by XREAL_1:7; then
   1 - |.(x-a)/b.|<=1+|.(x-a)/b.|-|.(x-a)/b.| by XREAL_1:9;
   hence thesis by XXREAL_0:28,XXREAL_0:25;
  end;
  then
  0 <= f.x & f.x <= 1 by B2;
  hence thesis by B3;
 end; then
 f is [.0,1.] -valued;
 hence thesis;
end;
