reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;

theorem
  for a, b, c, d being Real st a <= b & c <= d holds |.b-a.| +
  |.d-c.| = (b-a) + (d-c)
proof
  let a, b, c, d be Real;
  assume that
A1: a <= b and
A2: c <= d;
  a - a <= b - a by A1,XREAL_1:13;
  then
A3: |.b-a.| = b - a by ABSVALUE:def 1;
  c - c <= d - c by A2,XREAL_1:13;
  hence thesis by A3,ABSVALUE:def 1;
end;
