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

theorem
  -M2 is_less_than M1 implies M1+M2 is Positive
proof
A1: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A2: Indices M2=[:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices (-M2)=[:Seg n, Seg n:] by MATRIX_0:24;
A4: Indices (M1+M2)=[:Seg n, Seg n:] by MATRIX_0:24;
  assume
A5: -M2 is_less_than M1;
  for i,j st [i,j] in Indices (M1+M2) holds (M1+M2)*(i,j)>0
  proof
    let i,j;
    assume
A6: [i,j] in Indices (M1+M2);
    then (-M2)*(i,j)<M1*(i,j) by A5,A3,A4;
    then -M2*(i,j)<M1*(i,j) by A2,A4,A6,Th2;
    then M1*(i,j)+M2*(i,j)>0 by XREAL_1:62;
    hence thesis by A1,A4,A6,MATRIXR1:25;
  end;
  hence thesis;
end;
