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 Positive
let n be Nat; :: thesis: for M being Matrix of n,REAL st a > 0 & M is Positive holds
a * M is Positive
let M be Matrix of n,REAL; :: thesis: ( a > 0 & M is Positive implies a * M is Positive )
assume that
A1: a > 0 and
A2: M is Positive ; :: thesis: a * M is Positive
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 [i,j] in Indices (a * M) ; :: thesis: (a * M) * (i,j) > 0
then A3: [i,j] in Indices M by MATRIXR1:28;
then M * (i,j) > 0 by A2;
then a * (M * (i,j)) > 0 by A1, XREAL_1:129;
hence (a * M) * (i,j) > 0 by A3, Th4; :: thesis: verum

end;

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