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

theorem
  M is Nonpositive implies (-M) is Nonnegative
proof
A1: Indices M = [:Seg n, Seg n:] & Indices (-M) = [:Seg n, Seg n:] by
MATRIX_0:24;
  assume
A2: M is Nonpositive;
  for i,j st [i,j] in Indices (-M) holds (-M)*(i,j)>=0
  proof
    let i,j;
    assume
A3: [i,j] in Indices (-M);
    then M*(i,j)<=0 by A2,A1;
    then -M*(i,j)>=0;
    hence thesis by A1,A3,Th2;
  end;
  hence thesis;
end;
