let n be Nat; :: thesis: for M being Matrix of n, REAL st M is Nonnegative holds
- M is Nonpositive
let M be Matrix of n, REAL ; :: thesis: ( M is Nonnegative implies - M is Nonpositive )
assume A1: M is Nonnegative ; :: thesis: - M is Nonpositive
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, Def4;
then (- 1) * (M * i,j) <= 0 * (- 1) ;
then - (M * i,j) <= 0 ;
hence (- M) * i,j <= 0 by A2, A3, Th2; :: thesis: verum

end;

hence - M is Nonpositive by Def3; :: thesis: verum