
theorem MM60:
for f be Function of REAL,REAL, a,b,c be Real st
(for x be Real holds f.x= max(0,min(1, c*(1-|.(x-a)/b.|))) )
holds f is FuzzySet of REAL
proof
 let f be Function of REAL,REAL;
 let a,b,c be Real;
 assume A2: for x be Real holds f.x= max(0,min(1, c*(1-|.(x-a)/b.|)));
 ex g being Function of REAL,REAL st
 for x be Real holds g.x= c*(1-|.(x-a)/b.|)
 proof
  deffunc H1(Element of REAL) = In(c*(1-|.($1-a)/b.|),REAL);
  consider f being Function of REAL,REAL such that
  A1: for x being  Element of REAL holds f.x = H1(x) from FUNCT_2:sch 4;
  take f;
  for x be Real holds f.x= c*(1-|.(x-a)/b.|)
  proof
   let x be Real;
   reconsider x as Element of REAL by XREAL_0:def 1;
   f.x = H1(x) by A1;
   hence thesis;
  end;
  hence thesis;
 end; then
 consider g being Function of REAL,REAL such that
 A4:for x be Real holds g.x= c*(1-|.(x-a)/b.|);
 for x be Real holds f.x= max(0,min(1, g.x))
 proof
  let x be Real;
  f.x = max(0,min(1, c*(1-|.(x-a)/b.|))) by A2
     .= max(0,min(1, g.x)) by A4;
  hence thesis;
 end;
 hence thesis by MM40;
end;
