reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;

theorem Th49:
  idseq (i+1) = (idseq i) ^ <*i+1*>
proof
  set p = idseq (i+1);
  consider q being FinSequence , a being object such that
A1: p = q^<*a*> by FINSEQ_1:46;
A2: len p = i + 1 & len p = len q + 1 by A1,Th14,CARD_1:def 7;
A3: dom q = Seg len q by FINSEQ_1:def 3;
A4: for a being object st a in Seg i holds q.a = a
  proof
    i <= i+1 by NAT_1:11;
    then
A5: Seg i c= Seg (i+1) by FINSEQ_1:5;
    let a be object;
    assume
A6: a in Seg i;
    then ex j be Nat st a = j & 1 <= j & j <= i;
    then reconsider j = a as Nat;
    p.j = q.j by A1,A2,A3,A6,FINSEQ_1:def 7;
    hence thesis by A6,A5,FUNCT_1:18;
  end;
  p.(i+1) = i+1 by FINSEQ_1:4,FUNCT_1:18;
  then a = i+1 by A1,A2,FINSEQ_1:42;
  hence thesis by A1,A2,A3,A4,FUNCT_1:17;
end;
