reserve A for non empty closed_interval Subset of REAL;

theorem FU710a:
for a,b,c be Real st b > 0 & c > 0 holds
for x be Real st not x in ['a-c,a+c'] holds max(0, b - |. b*(x-a)/c .|) = 0
proof
 let a,b,c be Real;
 assume that
 A1: b > 0 and
 A2: c > 0;
  let x be Real;
  assume A4: not x in ['a-c,a+c'];
  deffunc H1(Element of REAL)
   = In(max(0,b - |. b*($1-a)/c .|),REAL);
  consider h being Function of REAL,REAL such that
  A5: for x being Element of REAL holds h.x = H1(x) from FUNCT_2:sch 4;
  A6: for x be Real holds h.x = max(0, b - |. b*(x-a)/c .|)
  proof
   let x be Real;
   reconsider x as Element of REAL by XREAL_0:def 1;
   h.x = In(max(0,b - |. b*(x-a)/c .|),REAL) by A5;
   hence thesis;
  end;
  hence 0 = h.x by A4,FUZZY_7:19,A1,A2
  .= max(0,b - |. b*(x-a)/c .|) by A6;
end;
