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

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