reserve i,j,k,n for Nat;
reserve D for non empty set,
  p for Element of D,
  f,g for FinSequence of D;

theorem
  len(Ins(f,n,p)) = len f + 1
proof
  per cases;
  suppose
A1: n <= len f;
    thus len Ins(f,n,p) = len((f|n)^<*p*>) + len(f/^n) by FINSEQ_1:22
      .= len((f|n)^<*p*>) + (len f - n) by A1,RFINSEQ:def 1
      .= len(f|n)+len<*p*> + (len f - n) by FINSEQ_1:22
      .= len(f|n)+1+(len f - n) by FINSEQ_1:39
      .= n+1+(len f - n) by A1,FINSEQ_1:59
      .= len f + 1;
  end;
  suppose
    len f < n;
    then Ins(f,n,p) = f^<*p*> by Th68;
    hence thesis by FINSEQ_2:16;
  end;
end;
