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+M4 implies M1-M3 is_less_or_equal_with M4-M2
proof
  assume
A1: M1+M2 is_less_or_equal_with M3+M4;
A2: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A3: width M2=width M4 by Lm3;
A4: Indices (M1+M2)=[:Seg n,Seg n:] by MATRIX_0:24;
A5: Indices M4=[:Seg n, Seg n:] & len M2=len M4 by Lm3,MATRIX_0:24;
A6: Indices M3=[:Seg n, Seg n:] by MATRIX_0:24;
A7: Indices (M1-M3)=[:Seg n,Seg n:] by MATRIX_0:24;
A8: len M1=len M3 & width M1=width M3 by Lm3;
  for i,j st [i,j] in Indices (M1-M3) holds (M1-M3)*(i,j)<=(M4-M2)*(i,j)
  proof
    let i,j;
    assume
A9: [i,j] in Indices (M1-M3);
    then (M1+M2)*(i,j)<=(M3+M4)*(i,j) by A1,A4,A7;
    then M1*(i,j)+M2*(i,j)<=(M3+M4)*(i,j) by A2,A7,A9,MATRIXR1:25;
    then M1*(i,j)+M2*(i,j)<=M3*(i,j)+M4*(i,j) by A6,A7,A9,MATRIXR1:25;
    then M1*(i,j)-M3*(i,j)<=M4*(i,j)-M2*(i,j) by XREAL_1:21;
    then (M1-M3)*(i,j)<=M4*(i,j)-M2*(i,j) by A2,A7,A8,A9,Th3;
    hence thesis by A7,A5,A3,A9,Th3;
  end;
  hence thesis;
end;
