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 Th5:
  for n being Nat, R being Element of n-tuples_on REAL
  holds mlt(n |-> (0 qua Real),R) = n |-> (0 qua Real)
proof
  let n be Nat, R be Element of n-tuples_on REAL;
A1: len(mlt(n |-> In(0,REAL),R)) =n by CARD_1:def 7;
A2: for k be Nat st 1 <= k & k <= len mlt(n |-> (0 qua Real),R) holds mlt(n
  |-> (0 qua Real),R).k = (n |-> (0 qua Real)).k
  proof
    let k be Nat;
    assume 1 <= k & k <= len mlt(n |-> (0 qua Real),R);
    mlt(n |-> (0 qua Real),R).k = (n |-> (0 qua Real)).k*R.k by RVSUM_1:60
      .=0*R.k;
    hence thesis;
  end;
  len (n |-> (0 qua Real)) = n by CARD_1:def 7;
  hence thesis by A1,A2,FINSEQ_1:14;
end;
