reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th33:
  for x,y being FinSequence of REAL st (for i st i in dom x holds
x.i >= 0) & (for i st i in dom y holds y.i >= 0) holds for k st k in dom mlt(x,
  y) holds (mlt(x,y)).k >= 0
proof
A1: for z being FinSequence of REAL st (for i st i in dom z holds z.i >= 0)
  holds for i holds z.i >=0
  proof
    let z be FinSequence of REAL such that
A2: for i st i in dom z holds z.i >=0;
    hereby
      let i;
      per cases;
      suppose
        not i in dom z;
        hence z.i >= 0 by FUNCT_1:def 2;
      end;
      suppose
        i in dom z;
        hence z.i >=0 by A2;
      end;
    end;
  end;
  let x,y be FinSequence of REAL such that
A3: ( for i st i in dom x holds x.i >= 0)& for i st i in dom y holds y.i >= 0;
  hereby
    let k;
    assume k in dom mlt(x,y);
A4: (mlt(x,y)).k = x.k * y.k by RVSUM_1:59;
    x.k >=0 & y.k >=0 by A3,A1;
    hence (mlt(x,y)).k >= 0 by A4;
  end;
end;
