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 implies M1-M2 is Positive
proof
  assume
A1: M1 is Positive & M2 is Negative;
A2: Indices M2 = [:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices M1 = [:Seg n, Seg n:] & Indices (M1-M2) = [: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 (M1-M2) holds (M1-M2)*(i,j)>0
  proof
    let i,j;
    assume
A5: [i,j] in Indices (M1-M2);
    then M1*(i,j)>0 & M2*(i,j)<0 by A1,A2,A3;
    then M1*(i,j)-M2*(i,j)>0-0;
    hence thesis by A3,A4,A5,Th3;
  end;
  hence thesis;
end;
