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

theorem
  M1 is_less_or_equal_with M2 & a<=0 implies a*M2 is_less_or_equal_with a*M1
proof
  assume that
A1: M1 is_less_or_equal_with M2 and
A2: a<=0;
A3: Indices (a*M2) = Indices M2 by MATRIXR1:28;
A4: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
  for i,j st [i,j] in Indices (a*M2) holds (a*M2)*(i,j)<=(a*M1)*(i,j)
  proof
    let i,j;
    assume
A5: [i,j] in Indices (a*M2);
    then
A6: [i,j] in Indices M1 by A4,MATRIX_0:24;
    then M1*(i,j)<=M2*(i,j) by A1;
    then a*(M2*(i,j))<=a*(M1*(i,j)) by A2,XREAL_1:65;
    then (a*M2)*(i,j)<=a*(M1*(i,j)) by A3,A5,Th4;
    hence thesis by A6,Th4;
  end;
  hence thesis;
end;
