reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th19:
  { 0*n } is linear_manifold
proof
  let x, y be Element of REAL n, a,b be Element of REAL;
  assume that
A1: x in {0*n} and
A2: y in {0*n};
  reconsider nn=n as Element of NAT by ORDINAL1:def 12;
A3: y=0*n by A2,TARSKI:def 1;
  x= 0*n by A1,TARSKI:def 1;
  then a*x+b*y =0*nn + b*(0*nn) by A3,EUCLID_4:2
    .=0*nn + (0*nn) by EUCLID_4:2
    .= 0*nn by EUCLID_4:1;
  hence a*x+b*y in { 0*n} by TARSKI:def 1;
end;
