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

theorem
  M2 is_less_than M1 implies M1-M2 is Positive
proof
  assume
A1: M2 is_less_than M1;
A2: Indices M2 = [:Seg n, Seg n:] by MATRIX_0:24;
A3: width M1=width M2 by Lm3;
A4: Indices (M1-M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A5: Indices M1 = [:Seg n, Seg n:] & len M1=len M2 by Lm3,MATRIX_0:24;
  for i,j st [i,j] in Indices (M1-M2) holds (M1-M2)*(i,j)>0
  proof
    let i,j;
    assume
A6: [i,j] in Indices (M1-M2);
    then M1*(i,j)>M2*(i,j) by A1,A2,A4;
    then M1*(i,j)-M2*(i,j)>0 by XREAL_1:50;
    hence thesis by A4,A5,A3,A6,Th3;
  end;
  hence thesis;
end;
