reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;
reserve A for Function of P, NAT;
reserve U, V, W for Subset of P*;
reserve k,l,m,n,i,j for Nat,
  a, b, c for object,
  x, y, z, X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R for FinSequence-membered set;
reserve B, C for antichain;
reserve S, T for Polish-language;
reserve A for Polish-arity-function of T;
reserve U, V, W for Polish-language of T;
reserve F, G for Polish-WFF of T, A;

theorem Th64:
  for S for p, q being Element of S for r being Element of S^^2 st r = p^q
    holds decomp(S, 2, r) = <*p, q*>
proof
  let S;
  let p, q be Element of S;
  let r be Element of S^^2;
  assume A1: r = p^q;
  set w = decomp(S, 2, r);
  1 in Seg 2 by FINSEQ_1:2, TARSKI:def 2;
  then consider k such that
    A2: 1 = k+1 and
    A3: w.1 = S-head((S^^k)-tail r) by Def32;
  A5: w.1 = S-head r by A2, A3, Th58
      .= p by A1;
  2 in Seg 2 by FINSEQ_1:2, TARSKI:def 2;
  then consider m such that
    A7: 2 = m+1 and
    A8: w.2 = S-head((S^^m)-tail r) by Def32;
  thus thesis by A1, A5, A7, A8, Th62, FINSEQ_1:44;
end;
