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 & M3 is_less_than M4 implies M1-M4
  is_less_than M2-M3
proof
  assume
A1: M1 is_less_or_equal_with M2 & M3 is_less_than M4;
A2: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices M2 = [:Seg n, Seg n:] & len M2=len M3 by Lm3,MATRIX_0:24;
A4: Indices M3 = [:Seg n, Seg n:] by MATRIX_0:24;
A5: width M2=width M3 by Lm3;
A6: Indices (M1-M4) = [:Seg n, Seg n:] by MATRIX_0:24;
A7: len M1=len M4 & width M1=width M4 by Lm3;
  for i,j st [i,j] in Indices (M1-M4) holds (M1-M4)*(i,j)<(M2-M3)*(i,j)
  proof
    let i,j;
    assume
A8: [i,j] in Indices (M1-M4);
    then M1*(i,j)<=M2*(i,j) & M3*(i,j)<M4*(i,j) by A1,A2,A4,A6;
    then M1*(i,j)-M4*(i,j)<M2*(i,j)-M3*(i,j) by XREAL_1:15;
    then (M1-M4)*(i,j)<M2*(i,j)-M3*(i,j) by A2,A6,A7,A8,Th3;
    hence thesis by A6,A3,A5,A8,Th3;
  end;
  hence thesis;
end;
