reserve A for non empty closed_interval Subset of REAL;

theorem FU710:
for a,b,c,d be Real, f be Function of REAL,REAL st
b > 0 & c > 0 & d > 0 &
(for x be Real holds f.x = min(d,max(0, b - |. b*(x-a)/c .|))) holds
for x be Real st not x in ['a-c,a+c'] holds f.x = 0
proof
 let a,b,c,d be Real, f be Function of REAL,REAL;
 assume that
 A1: b > 0 & c > 0 and
 A2: d > 0 and
 A3: for x be Real holds f.x = min(d, max(0, b - |. b*(x-a)/c .|));
  let x be Real;
  assume A4: not x in ['a-c,a+c'];
  f. x = min(d, max(0, b - |. b*(x-a)/c .|)) by A3
  .= 0 by FU710b,A1,A2,A4;
  hence thesis;
end;
