reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem
  p | X is FinSequence iff ex k being Element of NAT st X /\ dom p = Seg k
proof
  thus p | X is FinSequence implies
    ex k being Element of NAT st X /\ dom p = Seg k
  proof
    assume p | X is FinSequence;
    then consider k being Nat such that
A1: dom(p | X) = Seg k by FINSEQ_1:def 2;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    take k;
    thus thesis by A1,RELAT_1:61;
  end;
  given k being Element of NAT such that
A2: X /\ dom p = Seg k;
  dom(p | X) = Seg k by A2,RELAT_1:61;
  hence thesis by FINSEQ_1:def 2;
end;
