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_0:24;

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

- 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_0:24;

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

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

then (- 1) * (M * (i,j)) < 0 * (- 1) by XREAL_1:69;

then - (M * (i,j)) < 0 ;

hence (- M) * (i,j) < 0 by A1, A3, Th2; :: thesis: verum

end;assume A3: [i,j] in Indices (- M) ; :: thesis: (- M) * (i,j) < 0

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 (- M) * (i,j) < 0 by A1, A3, Th2; :: thesis: verum