let a be Element of REAL ; :: thesis: for n being Nat
for M being Matrix of n, REAL st a >= 0 & M is Nonpositive holds
a * M is Nonpositive
let n be Nat; :: thesis: for M being Matrix of n, REAL st a >= 0 & M is Nonpositive holds
a * M is Nonpositive
let M be Matrix of n, REAL ; :: thesis: ( a >= 0 & M is Nonpositive implies a * M is Nonpositive )
assume A1: ( a >= 0 & M is Nonpositive ) ; :: thesis: a * M is Nonpositive
A2: ( len (a * M) = len M & width (a * M) = width M ) by MATRIXR1:27;
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 A1, A3, Def3;
then a * (M * i,j) <= 0 by A1;
hence (a * M) * i,j <= 0 by A2, A3, A4, Th4; :: thesis: verum

end;

hence a * M is Nonpositive by Def3; :: thesis: verum