let n be Nat; :: thesis: for M being Matrix of n, REAL st M is Negative holds
- M is Positive
let M be Matrix of n, REAL ; :: thesis: ( M is Negative implies - M is Positive )
assume A1: M is Negative ; :: thesis: - M is Positive
A2: ( Indices M = [:(Seg n),(Seg n):] & Indices (- M) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
for i, j being Nat st [i,j] in Indices (- M) holds
(- M) * i,j > 0

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (- M) implies (- M) * i,j > 0 )
assume A3: [i,j] in Indices (- M) ; :: thesis: (- M) * i,j > 0
then M * i,j < 0 by A1, A2, Def2;
then - (M * i,j) > 0 by XREAL_1:60;
hence (- M) * i,j > 0 by A2, A3, Th2; :: thesis: verum

end;

hence - M is Positive by Def1; :: thesis: verum