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

theorem
  M1+M2 is_less_or_equal_with M3 implies M1 is_less_or_equal_with M3-M2
proof
  assume
A1: M1+M2 is_less_or_equal_with M3;
A2: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
A3: width M2=width M3 by Lm3;
A4: Indices (M1+M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A5: Indices M3 = [:Seg n, Seg n:] & len M2=len M3 by Lm3,MATRIX_0:24;
  for i,j st [i,j] in Indices M1 holds M1*(i,j)<=(M3-M2)*(i,j)
  proof
    let i,j;
    assume
A6: [i,j] in Indices M1;
    then (M1+M2)*(i,j)<=M3*(i,j) by A1,A2,A4;
    then M1*(i,j)+M2*(i,j)<=M3*(i,j) by A6,MATRIXR1:25;
    then M1*(i,j)<=M3*(i,j)-M2*(i,j) by XREAL_1:19;
    hence thesis by A2,A5,A3,A6,Th3;
  end;
  hence thesis;
end;
