reserve A for non empty closed_interval Subset of REAL;

theorem FU710b:
for a,b,c,d be Real st b > 0 & c > 0 & d > 0
holds
for x be Real st not x in ['a-c,a+c'] holds
min(d,max(0, b - |. b*(x-a)/c .|)) = 0
proof
 let a,b,c,d be Real;
 assume that
 A1: b > 0 and
 A2: c > 0 and
 A3: d > 0;
  let x be Real;
  assume not x in ['a-c,a+c']; then
  min(d,max(0, b - |. b*(x-a)/c .|)) = min(d,0) by FU710a,A1,A2
  .= 0 by XXREAL_0:def 9,A3;
  hence thesis;
end;
