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+M4 is_less_or_equal_with M3-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: Indices (M1+M2)=[:Seg n,Seg n:] by MATRIX_0:24;
A4: Indices (M1+M4)=[:Seg n,Seg n:] by MATRIX_0:24;
A5: len M2=len M3 & width M2=width M3 by Lm3;
A6: Indices M3=[:Seg n, Seg n:] by MATRIX_0:24;
A7: len M3=len M4 & width M3=width M4 by Lm3;
  for i,j st [i,j] in Indices (M1+M4) holds (M1+M4)*(i,j)<=(M3-M2)*(i,j)
  proof
    let i,j;
    assume
A8: [i,j] in Indices (M1+M4);
    then (M1+M2)*(i,j)<=(M3-M4)*(i,j) by A1,A3,A4;
    then M1*(i,j)+M2*(i,j)<=(M3-M4)*(i,j) by A2,A4,A8,MATRIXR1:25;
    then M1*(i,j)+M2*(i,j)<=M3*(i,j)-M4*(i,j) by A6,A4,A7,A8,Th3;
    then M1*(i,j)+M4*(i,j)<=M3*(i,j)-M2*(i,j) by XREAL_1:22;
    then (M1+M4)*(i,j)<=M3*(i,j)-M2*(i,j) by A2,A4,A8,MATRIXR1:25;
    hence thesis by A6,A4,A5,A8,Th3;
  end;
  hence thesis;
end;
