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

theorem
  M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| implies
  M1+M2 is Negative
proof
  assume that
A1: M1 is Positive and
A2: M2 is Negative and
A3: |:M1:| is_less_than |:M2:|;
A4: Indices M1 = [:Seg n, Seg n:] by MATRIX_0:24;
A5: Indices (M1+M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A6: Indices M2 = [:Seg n, Seg n:] by MATRIX_0:24;
  for i,j st [i,j] in Indices (M1+M2) holds (M1+M2)*(i,j)<0
  proof
    let i,j;
    assume
A7: [i,j] in Indices (M1+M2);
    then [i,j] in Indices |:M1:| by A4,A5,Th5;
    then |:M1:|*(i,j)<|:M2:|*(i,j) by A3;
    then |.M1*(i,j).|<|:M2:|*(i,j) by A4,A5,A7,Def7;
    then |.M1*(i,j).|<|.M2*(i,j).| by A6,A5,A7,Def7;
    then
A8: |.M1*(i,j).|-|.M2*(i,j).|<|.M2*(i,j).|-|.M2*(i,j).| by XREAL_1:9;
    M2*(i,j)<0 by A2,A6,A5,A7;
    then
A9: -(M2*(i,j))=|.M2*(i,j).| by ABSVALUE:def 1;
    M1*(i,j)>0 by A1,A4,A5,A7;
    then |.M1*(i,j).|=M1*(i,j) by ABSVALUE:def 1;
    then M1*(i,j)+M2*(i,j)=|.M1*(i,j).|-|.M2*(i,j).| by A9;
    hence thesis by A4,A5,A7,A8,MATRIXR1:25;
  end;
  hence thesis;
end;
