reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;

theorem Th03:
  for a,b,c,d being object st (a = x.1 or a = x.2 or a = x.3 or a = x.4) &
  (b = x.1 or b = x.2 or b = x.3 or b = x.4) &
  (c = x.1 or c = x.2 or c = x.3 or c = x.4) &
  (d = x.1 or d = x.2 or d = x.3 or d = x.4) holds
  <* a,b,c,d *> is Tuple of 4,X
  proof
    let a,b,c,d be object;
    assume
A1: (a = x.1 or a = x.2 or a = x.3 or a = x.4) &
      (b = x.1 or b = x.2 or b = x.3 or b = x.4) &
      (c = x.1 or c = x.2 or c = x.3 or c = x.4) &
      (d = x.1 or d = x.2 or d = x.3 or d = x.4);
      set y = <* a,b,c,d *>;
    dom x = Seg 4 by FINSEQ_2:124; then
A2: 1 in dom x & 2 in dom x & 3 in dom x & 4 in dom x;
    reconsider d19 = y.1, d29 = y.2, d39 = y.3,
      d49 = y.4 as Element of X by A1,A2,FINSEQ_2:11;
    a is Element of X & b is Element of X & c is Element of X &
      d is Element of X by A1,A2,FINSEQ_2:11; then
    y in the set of all <* d1, d2, d3, d4 *> where
      d1,d2,d3,d4 is Element of X;
    then y in 4-tuples_on X by Th02;
    hence thesis by FINSEQ_2:131;
  end;
