reserve i, n for Nat,
  x, y, a for Real,
  v for Element of n-tuples_on REAL,
  p, p1, p2, p3, q, q1, q2 for Point of TOP-REAL n;

theorem Th2:
  mlt(v, 0*n) = 0*n
proof
  len 0*n = n by CARD_1:def 7;
  then reconsider z= 0*n as Element of n-tuples_on REAL by FINSEQ_2:92;
A1: len 0*n = n by CARD_1:def 7;
A2: for j be Nat st j in dom 0*n holds mlt(v,0*n).j=(0*n).j
  proof
    let j be Nat;
    assume
 j in dom 0*n;
    thus mlt(v,0*n).j = 0 by Th1
      .= (n |-> 0).j
      .= (0*n).j by EUCLID:def 4;
  end;
  len mlt(v, z) = n by CARD_1:def 7;
  then dom mlt(v, 0*n) = dom 0*n by A1,FINSEQ_3:29;
  hence thesis by A2,FINSEQ_1:13;
end;
