
theorem Th7:
  for n be Element of NAT for p,q be Element of n-tuples_on NAT
  holds p <= q or p > q
proof
  let n be Element of NAT;
  let p,q be Element of n-tuples_on NAT;
  assume
A1: not p <= q;
  then consider i be Element of NAT such that
A2: i in Seg n and
A3: p.i <> q.i and
A4: for k be Nat st 1 <= k & k < i holds p.k = q.k by Th6;
  take i;
  thus i in Seg n by A2;
  not p < q by A1;
  then p.i >= q.i by A2,A4;
  hence q.i < p.i by A3,XXREAL_0:1;
  thus thesis by A4;
end;
