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

theorem
  M1-M2 is_less_or_equal_with M3-M4 implies M4-M2 is_less_or_equal_with M3-M1
proof
A1: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
A2: Indices M3 = [:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices (M1-M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A4: Indices M4 = [:Seg n, Seg n:] by MATRIX_0:24;
A5: Indices (M4-M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A6: len M1=len M3 & width M1=width M3 by Lm3;
A7: len M3=len M4 & width M3=width M4 by Lm3;
A8: len M1=len M2 & width M1=width M2 by Lm3;
  assume
A9: M1-M2 is_less_or_equal_with M3-M4;
  for i,j st [i,j] in Indices (M4-M2) holds (M4-M2)*(i,j)<=(M3-M1)*(i,j)
  proof
    let i,j;
    assume
A10: [i,j] in Indices (M4-M2);
    then (M1-M2)*(i,j)<=(M3-M4)*(i,j) by A9,A3,A5;
    then M1*(i,j)-M2*(i,j)<=(M3-M4)*(i,j) by A1,A5,A8,A10,Th3;
    then M1*(i,j)-M2*(i,j)<=M3*(i,j)-M4*(i,j) by A2,A5,A7,A10,Th3;
    then M4*(i,j)-M2*(i,j)<=M3*(i,j)-M1*(i,j) by XREAL_1:17;
    then (M4-M2)*(i,j)<=M3*(i,j)-M1*(i,j) by A4,A5,A8,A7,A6,A10,Th3;
    hence thesis by A2,A5,A6,A10,Th3;
  end;
  hence thesis;
end;
