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

theorem
  M1 is Nonnegative & M2 is_less_than M3 implies M2-M1 is_less_than M3
proof
  assume
A1: M1 is Nonnegative & M2 is_less_than M3;
A2: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices M2=[:Seg n, Seg n:] & Indices (M2-M1)=[:Seg n, Seg n:] by
MATRIX_0:24;
A4: len M1=len M2 & width M1=width M2 by Lm3;
  for i,j st [i,j] in Indices (M2-M1) holds (M2-M1)*(i,j)<M3*(i,j)
  proof
    let i,j;
    assume
A5: [i,j] in Indices (M2-M1);
    then M1*(i,j)>=0 & M2*(i,j)<M3*(i,j) by A1,A2,A3;
    then M2*(i,j)-M1*(i,j)<M3*(i,j) by XREAL_1:51;
    hence thesis by A3,A4,A5,Th3;
  end;
  hence thesis;
end;
