reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th10:
  p is nonnegative & r>=0 implies r*p is nonnegative
proof
  assume that
A1: p is nonnegative and
A2: r>=0;
  now
    let k;
    assume k in dom (r*p);
    then k in dom p by VALUED_1:def 5;
    then
A3: p.k >= 0 by A1;
    (r*p).k = r*(p.k) by RVSUM_1:44;
    hence (r*p).k >= 0 by A2,A3;
  end;
  hence thesis;
end;
