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 Th1:
  mlt(v, 0*n).i = 0
proof
A1: 0*n = n |-> (0 qua set) by EUCLID:def 4;
  set v0 = (0*n).i;
A2: dom(0*n) = Seg n by FINSEQ_2:124;
  i in Seg n or not i in Seg n;
  then
A3: v0=0 by A1,A2,FUNCOP_1:7,FUNCT_1:def 2;
  mlt(v,0*n).i = v.i * v0 by RVSUM_1:60
    .= 0 by A3;
  hence thesis;
end;
