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
  for D being non empty set for p being FinSequence of D holds
  Del(p,i) is FinSequence of D
proof
  let D be non empty set, p be FinSequence of D;
  per cases;
  suppose
    i in dom p;
    then reconsider D9=Seg(len p) as non empty set by FINSEQ_1:def 3;
    for x being object holds x in (Seg(len p) \ {i}) implies x in Seg(len p)
    by XBOOLE_0:def 5;
    then Seg(len p) \ {i} c= Seg(len p);
    then rng Sgm(Seg(len p) \ {i}) c= Seg(len p) by FINSEQ_1:def 14;
    then reconsider q=Sgm(Seg(len p) \ {i}) as FinSequence of D9
    by FINSEQ_1:def 4;
    p * q = Del(p,i) by FINSEQ_1:def 3;
    hence thesis by FINSEQ_2:31;
  end;
  suppose
    not i in dom p;
    hence thesis by Th102;
  end;
end;
