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 Th60:
  for S, T, n for s being Element of S^^n, t being Element of T^^n
      st S c= T & s = t holds decomp( S, n, s ) = decomp( T, n, t )
proof
  let S, T, n;
  let s be Element of S^^n;
  let t be Element of T^^n;
  assume that A1: S c= T and A2: s = t;
  set p = decomp( S, n, s );
  set q = decomp( T, n, t );
  A4: dom p = Seg n & dom q = Seg n by Def32;
  for a st a in Seg n holds p.a = q.a
    proof
    let a;
    assume A6: a in Seg n;
    then reconsider a as Nat;
    consider k such that A7: a = k+1 and A8: p.a = S-head((S^^k)-tail s)
        by A6, Def32;
    consider l such that A9: a = l+1 and A10: q.a = T-head((T^^l)-tail t)
        by A6, Def32;
    A11: S^^l c= T^^l by A1, Th17;
    l+1 <= n by A6, A9, FINSEQ_1:1;
    then A12: s is (S^^l)-headed & (S^^l)-tail s is S-headed by Th57;
    then A14: (T^^l)-tail t is S-headed by A2, A11, Th55;
    (S^^k)-tail s = (T^^l)-tail t by A2, A7, A9, A11, A12, Th55;
    hence thesis by A1, A8, A10, A14, Th55;
    end;
  hence thesis by A4, FUNCT_1:def 11;
end;
