 reserve A for non empty Subset of REAL;

theorem
  for a,b,c be Real, f be Function of REAL,REAL st
    b > 0 & c > 0 &
  ( for x be Real holds f.x = max(0, b - |. b*(x-a)/c .|) ) holds
  f = b (#) (TriangularFS (a-c,a,a+c))
proof
 let a,b,c be Real, f be Function of REAL,REAL;
 assume that
 A1: b > 0 and
 A2: c > 0 and
 A3: for x be Real holds f.x = max(0, b - |. b*(x-a)/c .|);
 set g = b (#) (TriangularFS (a-c,a,a+c));
 A4:  dom f = REAL by FUNCT_2:def 1
  .= dom (b (#) (TriangularFS (a-c,a,a+c))) by FUNCT_2:def 1;
 for x being object st x in dom f holds f . x = g . x
 proof
  let x be object;
  assume x in dom f; then
  reconsider x as Real;
  ( b (#) (TriangularFS (a-c,a,a+c)) ) .x
   = b * (TriangularFS (a-c,a,a+c)) .x by VALUED_1:6
  .= b * max(0,1-|.(x-a)/c.|) by FUZZY_5:65,A2
  .= max(b*0,b*(1-|.(x-a)/c.|)) by FUZZY_2:41,A1
  .= max(0,(b - b*|.(x-a)/c.|))
  .= max(0,(b - |.b.| * |.(x-a)/c.|)) by A1,ABSVALUE:def 1
  .= max(0,(b - |.b*((x-a)/c).|)) by COMPLEX1:65
  .= max(0,(b - |.b*(x-a)/c.|)) by XCMPLX_1:74
  .= f.x by A3;
  hence thesis;
 end;
 hence thesis by A4,FUNCT_1:def 11;
end;
