
theorem
  for D be non empty set, x be Element of 4-tuples_on(4-tuples_on D),
  k be Element of NAT st k in Seg 4 holds
  ex x1,x2,x3,x4 be Element of D st x1 = (x.k).1 & x2 = (x.k).2 &
  x3 = (x.k).3 & x4 = (x.k).4
proof
  let D be non empty set, x be Element of 4-tuples_on(4-tuples_on D),
  k be Element of NAT;
  assume
AS: k in Seg 4;
  x in 4-tuples_on(4-tuples_on D);
  then ex s be Element of (4-tuples_on D)* st x = s & len s = 4;
  then k in dom x by AS,FINSEQ_1:def 3;
  then x.k in rng x by FUNCT_1:3;
  then x.k in 4-tuples_on (D);
  then
Q13: ex s be Element of D* st x.k = s & len s = 4;
  then reconsider xk = x.k as Element of D*;
  1 in Seg 4;
  then 1 in dom xk by Q13,FINSEQ_1:def 3;
  then xk.1 in rng xk by FUNCT_1:3;
  then reconsider x1 = xk.1 as Element of D;
  2 in Seg 4;
  then 2 in dom xk by Q13,FINSEQ_1:def 3;
  then xk.2 in rng xk by FUNCT_1:3;
  then reconsider x2 = xk.2 as Element of D;
  3 in Seg 4;
  then 3 in dom xk by Q13,FINSEQ_1:def 3;
  then xk.3 in rng xk by FUNCT_1:3;
  then reconsider x3 = xk.3 as Element of D;
  4 in Seg 4;
  then 4 in dom xk by Q13,FINSEQ_1:def 3;
  then xk.4 in rng xk by FUNCT_1:3;
  then reconsider x4 = xk.4 as Element of D;
  take x1,x2,x3,x4;
  thus thesis;
end;
