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

theorem
  M is Positive implies (-M) is Negative
proof
A1: Indices M = [:Seg n, Seg n:] & Indices -M = [:Seg n, Seg n:] by MATRIX_0:24
;
  assume
A2: M is Positive;
  for i,j st [i,j] in Indices (-M) holds (-M)*(i,j)<0
  proof
    let i,j;
    assume
A3: [i,j] in Indices (-M);
    then M*(i,j)>0 by A2,A1;
    then (-1)*(M*(i,j))<0*(-1) by XREAL_1:69;
    then -(M*(i,j))<0;
    hence thesis by A1,A3,Th2;
  end;
  hence thesis;
end;
