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 & M2 is_less_than M3 implies M1 is_less_than M3
proof
A1: Indices M1 = [:Seg n, Seg n:] & Indices M2 = [:Seg n, Seg n:] by
MATRIX_0:24;
  assume
A2: M1 is_less_than M2 & M2 is_less_than M3;
  for i,j st [i,j] in Indices M1 holds M1*(i,j)<M3*(i,j)
  proof
    let i,j;
    assume [i,j] in Indices M1;
    then M1*(i,j)<M2*(i,j) & M2*(i,j)<M3*(i,j) by A2,A1;
    hence thesis by XXREAL_0:2;
  end;
  hence thesis;
end;
