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

theorem
  M1 is Nonpositive & M3 is_less_than M2 implies M3+M1 is_less_than M2
proof
A1: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A2: Indices M3=[:Seg n, Seg n:] & Indices (M3+M1)=[:Seg n, Seg n:] by
MATRIX_0:24;
  assume
A3: M1 is Nonpositive & M3 is_less_than M2;
  for i,j st [i,j] in Indices (M3+M1) holds (M3+M1)*(i,j)<M2*(i,j)
  proof
    let i,j;
    assume
A4: [i,j] in Indices (M3+M1);
    then M1*(i,j)<=0 & M3*(i,j)<M2*(i,j) by A3,A1,A2;
    then M3*(i,j)+M1*(i,j)<M2*(i,j) by XREAL_1:36;
    hence thesis by A2,A4,MATRIXR1:25;
  end;
  hence thesis;
end;
