let n be Nat; :: thesis: for M being Matrix of n, REAL st M is Positive holds
- M is Negative
let M be Matrix of n, REAL ; :: thesis: ( M is Positive implies - M is Negative )
A1: ( Indices M = [:(Seg n),(Seg n):] & Indices (- M) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
assume A2: M is Positive ; :: thesis: - M is Negative
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 A2, A1, Def1;
then (- 1) * (M * (i,j)) < 0 * (- 1) by XREAL_1:71;
then - (M * (i,j)) < 0 ;
hence (- M) * (i,j) < 0 by A1, A3, Th2; :: thesis: verum

end;

hence - M is Negative by Def2; :: thesis: verum