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

theorem
  M1 is Nonpositive & M2 is_less_than M3 implies M2 is_less_than M3-M1
proof
  assume
A1: M1 is Nonpositive & M2 is_less_than M3;
A2: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A3: width M2=width M3 by Lm3;
A4: width M1=width M2 & len M2=len M3 by Lm3;
A5: Indices M2=[:Seg n, Seg n:] by MATRIX_0:24;
A6: Indices M3=[:Seg n, Seg n:] & len M1=len M2 by Lm3,MATRIX_0:24;
  for i,j st [i,j] in Indices M2 holds M2*(i,j)<(M3-M1)*(i,j)
  proof
    let i,j;
    assume
A7: [i,j] in Indices M2;
    then M1*(i,j)<=0 & M2*(i,j)<M3*(i,j) by A1,A2,A5;
    then M2*(i,j)<M3*(i,j)-M1*(i,j) by XREAL_1:53;
    hence thesis by A5,A6,A4,A3,A7,Th3;
  end;
  hence thesis;
end;
