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

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

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