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 & a>0 implies a*M1 is_less_than a*M2
proof
  assume that
A1: M1 is_less_than M2 and
A2: a>0;
A3: Indices (a*M1) = Indices M1 by MATRIXR1:28;
A4: Indices M2 = [:Seg n, Seg n:] by MATRIX_0:24;
  for i,j st [i,j] in Indices (a*M1) holds (a*M1)*(i,j)<(a*M2)*(i,j)
  proof
    let i,j;
    assume
A5: [i,j] in Indices (a*M1);
    then M1*(i,j)<M2*(i,j) by A1,A3;
    then a*(M1*(i,j))<a*(M2*(i,j)) by A2,XREAL_1:68;
    then
A6: (a*M1)*(i,j)<a*(M2*(i,j)) by A3,A5,Th4;
    [i,j] in Indices M2 by A4,A5,MATRIX_0:24;
    hence thesis by A6,Th4;
  end;
  hence thesis;
end;
