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-M2 is Negative
proof
  assume
A1: M1 is_less_than M2;
A2: Indices M1 = [:Seg n, Seg n:] & Indices (M1-M2) = [:Seg n, Seg n:] by
MATRIX_0:24;
A3: len M1=len M2 & width M1=width M2 by Lm3;
  for i,j st [i,j] in Indices (M1-M2) holds (M1-M2)*(i,j)<0
  proof
    let i,j;
    assume
A4: [i,j] in Indices (M1-M2);
    then M1*(i,j)<M2*(i,j) by A1,A2;
    then M1*(i,j)-M2*(i,j)<0 by XREAL_1:49;
    hence thesis by A2,A3,A4,Th3;
  end;
  hence thesis;
end;
