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

theorem
  M1 is_less_than M2 implies M1+M3 is_less_than M2+M3
proof
A1: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
A2: Indices M2 = [:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices (M1+M3) = [:Seg n, Seg n:] by MATRIX_0:24;
  assume
A4: M1 is_less_than M2;
  for i,j st [i,j] in Indices (M1+M3) holds (M1+M3)*(i,j)<(M2+M3)*(i,j)
  proof
    let i,j;
    assume
A5: [i,j] in Indices (M1+M3);
    then M1*(i,j)<M2*(i,j) by A4,A1,A3;
    then M1*(i,j)+M3*(i,j)<M2*(i,j)+M3*(i,j) by XREAL_1:8;
    then (M1+M3)*(i,j)<M2*(i,j)+M3*(i,j) by A1,A3,A5,MATRIXR1:25;
    hence thesis by A2,A3,A5,MATRIXR1:25;
  end;
  hence thesis;
end;
