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 Th63:
  for S for q being Element of S^^1 holds decomp(S, 1, q) = <*q*>
proof
  let S;
  let q be Element of S^^1;
  set w = decomp(S, 1, q);
  1 in Seg 1 by FINSEQ_1:2, TARSKI:def 1;
  then consider k such that
    A2: 1 = k+1 and
    A3: w.1 = S-head((S^^k)-tail q) by Def32;
  w.1 = S-head q by A2, A3, Th58 .= q;
  hence thesis by Th62, FINSEQ_1:40;
end;
