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 Negative implies a*M is Positive
proof
  assume that
A1: a<0 and
A2: M is Negative;
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,XREAL_1:130;
    hence thesis by A3,A4,Th4;
  end;
  hence thesis;
end;
