reserve a,b for Real,
  i,j,n for Nat,
  M,M1,M2,M3,M4 for Matrix of n, REAL;

theorem
  |:M:| is Nonnegative
proof
  for i,j st [i,j] in Indices |:M:| holds |:M:|*(i,j)>=0
  proof
    let i,j;
    assume
A1: [i,j] in Indices |:M:|;
    Indices |:M:| = Indices M by Th5;
    then |:M:|*(i,j)=|.M*(i,j).| by A1,Def7;
    hence thesis by COMPLEX1:46;
  end;
  hence thesis;
end;
