reserve A for non empty closed_interval Subset of REAL;

theorem L724:
for a,b,c,d,e be Real, f be Function of REAL,REAL st
c <> 0 & for x be Real holds f.x = min(d, max(e, b - |. b*(x-a)/c .|))
holds f is Lipschitzian
proof
 let a,b,c,d,e be Real, f be Function of REAL,REAL;
 assume A2: c <> 0;
 assume A1:for x be Real holds f.x = min(d, max(e, b - |. b*(x-a)/c .|));
 per cases;
  suppose B1: b = 0;
   ex r being Real st
   ( 0 < r &
   ( for x1, x2 being Real st x1 in dom f & x2 in dom f holds
   |.f.x1 - f.x2.| <= r * |.x1 - x2.| ) )
   proof
    take 1;
    thus 0 < 1;
     let x1, x2 be Real;
     |.f.x1 - f.x2.|
      =|.min(d, max(e, b - |. b*(x1-a)/c .|)) - f.x2.| by A1
     .=|.min(d, max(e, b - |. b*(x1-a)/c .|))
      - min(d, max(e, b - |. b*(x2-a)/c .|)).| by A1
      .= 0 by COMPLEX1:44,B1;
     hence thesis by COMPLEX1:46;
   end;
   hence thesis;
  end;
  suppose b <> 0;
   hence thesis by A1,A2,L724p;
  end;
end;
