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

theorem
  (M1-M3) is_less_or_equal_with (M2-M3) implies M1 is_less_or_equal_with M2
proof
  assume
A1: (M1-M3) is_less_or_equal_with (M2-M3);
A2: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
A3: width M2=width M3 by Lm3;
A4: Indices M2 = [:Seg n, Seg n:] & len M2=len M3 by Lm3,MATRIX_0:24;
A5: Indices (M1-M3) = [:Seg n, Seg n:] by MATRIX_0:24;
A6: len M1=len M3 & width M1=width M3 by Lm3;
  for i,j st [i,j] in Indices M1 holds M1*(i,j)<=M2*(i,j)
  proof
    let i,j;
    assume
A7: [i,j] in Indices M1;
    then (M1-M3)*(i,j)<=(M2-M3)*(i,j) by A1,A2,A5;
    then M1*(i,j)-M3*(i,j)<=(M2-M3)*(i,j) by A6,A7,Th3;
    then M1*(i,j)-M3*(i,j)<=M2*(i,j)-M3*(i,j) by A2,A4,A3,A7,Th3;
    then (M1*(i,j)-M3*(i,j))+M3*(i,j)<=(M2*(i,j)-M3*(i,j))+M3*(i,j) by
XREAL_1:6;
    hence thesis;
  end;
  hence thesis;
end;
