theorem Th41:
  |.n .--> a.|.a = n & for b being Element of A st b <> a holds |.
  n .--> a.|.b = 0
proof
  defpred P[Nat] means |.In($1,NAT) .--> a.|.a = $1;
A1: 0.-->a = {} & {} = <*>A by FINSEQ_2:58;
A2: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat such that
A3: |.In(n,NAT) .--> a.|.a = n;
    thus |.In(n+1,NAT) .--> a.|.a
       = |.(In(n,NAT) .--> a)^<*a*>.|.a by FINSEQ_2:60
      .= (|.In(n,NAT) .--> a.|[*]chi a).a by Th39
      .= n+(chi a).a by A3,Th29
      .= n+1 by Th31;
  end;
  (A-->0).a = 0;
  then
A4: P[0] by A1,Th37;
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A2);
   then |.In(n,NAT) .--> a.|.a = n;
  hence |.n .--> a.|.a = n;
  let b be Element of A such that
A5: b <> a;
  defpred P[Nat] means |.In($1,NAT) .--> a.|.b = 0;
A6: for n being Nat st P[n] holds P[(n+1)]
  proof
    let n be Nat such that
A7: |.In(n,NAT) .--> a.|.b = 0;
    thus |.In(n+1,NAT) .--> a.|.b
       = |.(In(n,NAT) .--> a)^<*a*>.|.b by FINSEQ_2:60
      .= (|.In(n,NAT) .--> a.|[*]chi a).b by Th39
      .= 0+(chi a).b by A7,Th29
      .= 0 by A5,Th31;
  end;
  (A-->0).b = 0;
  then
A8: P[0] by A1,Th37;
  for n being Nat holds P[n] from NAT_1:sch 2(A8,A6);
   then P[n];
  hence thesis;
end;
