reserve x for set;
reserve k, l for Nat;
reserve p, q for FinSequence;
reserve R for Relation;
reserve p, q for RedSequence of R;
reserve E for set;
reserve s, t for XFinSequence;
reserve p, q for XFinSequence-yielding FinSequence;

theorem
  s ^+ (p +^ t) = (s ^+ p) +^ t
proof
A1: now
    let k be Nat;
    assume k in dom (s ^+ (p +^ t));
    then
A2: k in dom (p +^ t) by Def2;
    then
A3: k in dom p by Def3;
    then
A4: k in dom (s ^+ p) by Def2;
    thus (s ^+ (p +^ t)).k = s^((p +^ t).k) by A2,Def2
      .= s^((p.k)^t) by A3,Def3
      .= s^(p.k)^t by AFINSQ_1:27
      .= ((s ^+ p).k)^t by A3,Def2
      .= ((s ^+ p) +^ t).k by A4,Def3;
  end;
  dom (s ^+ (p +^ t)) = dom (p +^ t) by Def2
    .= dom p by Def3
    .= dom (s ^+ p) by Def2
    .= dom ((s ^+ p) +^ t) by Def3;
  hence thesis by A1,FINSEQ_1:13;
end;
