reserve A for non empty closed_interval Subset of REAL;

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