 reserve A for non empty Subset of REAL;

theorem Th14:
  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 is Lipschitzian
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 .|);
 for x be Real holds f.x = max(0,min(b,b*(1-|. (x-a)/c .|)))
 proof
  let x be Real;
  max(0,min(b,b*(1-|. (x-a)/c .|)))
   = max(0, min(b, (b*1-b*|. (x-a)/c .|) ))
  .= max(0, min(b, (b - |.b.| * |. (x-a)/c .|) )) by A1,ABSVALUE:def 1
  .= max(0, min(b,(b - |.b*((x-a)/c).|))) by COMPLEX1:65
  .= max(0,min(b,(b - |.b*(x-a)/c.|))) by XCMPLX_1:74
  .= max(0,(b - |.b*(x-a)/c.|)) by XXREAL_0:def 9,Lm5;
  hence thesis by A3;
 end;
 hence thesis by FUZZY_5:70,A2;
end;
