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 Nonnegative implies a*M is Nonpositive
proof
  assume that
A1: a<=0 and
A2: M is Nonnegative;
A3: Indices (a*M) = Indices M by MATRIXR1:28;
  for i,j st [i,j] in Indices (a*M) holds (a*M)*(i,j)<=0
  proof
    let i,j;
    assume
A4: [i,j] in Indices (a*M);
    then M*(i,j)>=0 by A2,A3;
    then a*(M*(i,j))<=0 by A1;
    hence thesis by A3,A4,Th4;
  end;
  hence thesis;
end;
