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

theorem
  a>=0 & M is Positive implies a*M is Nonnegative
proof
  assume that
A1: a>=0 and
A2: M is Positive;
  for i,j st [i,j] in Indices (a*M) holds (a*M)*(i,j)>=0
  proof
    let i,j;
    assume [i,j] in Indices (a*M);
    then
A3: [i,j] in Indices M by MATRIXR1:28;
    then M*(i,j)>0 by A2;
    then a*(M*(i,j))>=0 by A1;
    hence thesis by A3,Th4;
  end;
  hence thesis;
end;
