reserve A for non empty closed_interval Subset of REAL;

theorem
for f be Function of REAL,REAL, a,b,c,d,e be Real st
(for x be Real holds f.x = min(d, max(e, b - |. b*(x-a)/c .|)))
holds
(for y be Real holds f.(a-y) = f.(a+y))
proof
 let f be Function of REAL,REAL, a,b,c,d,e be Real;
 assume A2: for x be Real holds f.x = min(d, max(e, b - |. b*(x-a)/c .|));
  let y be Real;
   thus f.(a-y) = min(d, max(e, b - |. b*(a-y-a)/c .|)) by A2
  .= min(d, max(e,b - |. (-(y*b))*(1/c) .|)) by XCMPLX_1:99
  .= min(d, max(e,b - |. -((y*b)*(1/c)) .|))
  .= min(d, max(e,b - |. -((y*b)/c) .|)) by XCMPLX_1:99
  .= min(d, max(e,b - |. b*((a+y)-a)/c .|)) by COMPLEX1:52
  .= f.(a+y) by A2;
end;
