let a be Real; :: thesis: for n being Nat
for M being Matrix of n,REAL st a <= 0 & M is Positive holds
a * M is Nonpositive
let n be Nat; :: thesis: for M being Matrix of n,REAL st a <= 0 & M is Positive holds
a * M is Nonpositive
let M be Matrix of n,REAL; :: thesis: ( a <= 0 & M is Positive implies a * M is Nonpositive )
assume that
A1: a <= 0 and
A2: M is Positive ; :: thesis: a * M is Nonpositive
A3: Indices (a * M) = Indices M by MATRIXR1:28;
for i, j being Nat st [i,j] in Indices (a * M) holds
(a * M) * (i,j) <= 0

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (a * M) implies (a * M) * (i,j) <= 0 )
assume A4: [i,j] in Indices (a * M) ; :: thesis: (a * M) * (i,j) <= 0
then M * (i,j) > 0 by A2, A3;
then a * (M * (i,j)) <= 0 by A1;
hence (a * M) * (i,j) <= 0 by A3, A4, Th4; :: thesis: verum

end;

hence a * M is Nonpositive ; :: thesis: verum