
theorem
for r,s be Real, f be Function of REAL,REAL st
for x be Real holds f.x = max(r,min(s,x))
holds
f is Lipschitzian
proof
 let r,s be Real;
 let f be Function of REAL,REAL;
 assume A1:for x be Real holds f.x= max(r,min(s,x));
 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;
  for x1, x2 being Real st x1 in dom f & x2 in dom f holds
  |.((f . x1) - (f . x2)).| <= 1 * |.(x1 - x2).|
  proof
   let x1, x2 be Real;
   |.((f . x1) - (f . x2)).| = |.(max(r,min(s,x1)) - (f . x2)).| by A1
   .=|.max(r,min(s,x1)) - max(r,min(s,x2)).| by A1;
   hence thesis by LeMM01;
  end;
  hence thesis;
 end;
 hence thesis;
end;
