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

theorem
  M1 is Negative & M2 is Negative implies M1+M2 is Negative
proof
  assume that
A1: M1 is Negative and
A2: M2 is Negative;
A3: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
A4: Indices (M1+M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A5: Indices M2 = [:Seg n, Seg n:] by MATRIX_0:24;
  for i,j st [i,j] in Indices (M1+M2) holds (M1+M2)*(i,j)<0
  proof
    let i,j;
    assume
A6: [i,j] in Indices (M1+M2);
    then M1*(i,j) < 0 by A1,A3,A4;
    then M1*(i,j)+M2*(i,j)<M2*(i,j) by XREAL_1:30;
    then M1*(i,j)+M2*(i,j)<0 by A2,A5,A4,A6;
    hence thesis by A3,A4,A6,MATRIXR1:25;
  end;
  hence thesis;
end;
