reserve A for non empty closed_interval Subset of REAL;

theorem F51:
  for a, b, c, d being Real holds
    |. min (c,max (d,a)) - min (c,max (d,b)) .| <= |. a - b .|
proof
  let a, b, c, d be Real;
  A1: |. min (c,max (d,a)) - min (c,max (d,b)) .|
    <= |. max (d,a) - max (d,b) .| by F51min;
  |. max (d,a) - max (d,b) .| <= |. a - b .| by F51max;
  hence thesis by A1,XXREAL_0:2;
end;
