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 M3-M2 is_less_than M3-M1
proof
  assume
A1: M1 is_less_than M2;
A2: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices (M3-M1) = [:Seg n, Seg n:] by MATRIX_0:24;
A4: width M2=width M3 by Lm3;
A5: Indices M3 = [:Seg n, Seg n:] & len M2=len M3 by Lm3,MATRIX_0:24;
A6: len M1=len M2 & width M1=width M2 by Lm3;
A7: for i,j st [i,j] in Indices (M3-M1) holds (M3-M2)*(i,j)<(M3-M1)*(i,j)
  proof
    let i,j;
    assume
A8: [i,j] in Indices (M3-M1);
    then M1*(i,j)<M2*(i,j) by A1,A2,A3;
    then M3*(i,j)-M2*(i,j)<M3*(i,j)-M1*(i,j) by XREAL_1:15;
    then (M3-M2)*(i,j)<M3*(i,j)-M1*(i,j) by A3,A5,A4,A8,Th3;
    hence thesis by A3,A6,A5,A4,A8,Th3;
  end;
  Indices (M3-M2) = [:Seg n, Seg n:] by MATRIX_0:24;
  hence thesis by A3,A7;
end;
