reserve a,b,p,r,r1,r2,s,s1,s2,x0,x for Real;
reserve f,g for PartFunc of REAL,REAL;
reserve X,Y for set;

theorem Th4:
  for n being Nat, R being Element of n-tuples_on REAL
st (for i being Element of NAT st i in dom R holds 0 = R.i) holds R = n |-> (0
  qua Real)
proof
  let n be Nat, R be Element of n-tuples_on REAL such that
A1: for i being Element of NAT st i in dom R holds 0 = R.i;
A2: for k be Nat st 1 <= k & k <= len R holds R.k = (n |-> 0).k
  proof
    let k be Nat;
    assume 1 <= k & k <= len R;
    then
 k in Seg len R by FINSEQ_1:1;
    then k in dom R by FINSEQ_1:def 3;
    then
A3: R.k = 0 by A1;
    thus thesis by A3;
  end;
  len R = n by CARD_1:def 7
    .= len (n |-> 0) by CARD_1:def 7;
  hence thesis by A2,FINSEQ_1:14;
end;
