for i, j being Nat st [i,j] in Indices (M @) holds

(M @) * (i,j) > 0

(M @) * (i,j) > 0

proof

hence
M @ is Positive
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (M @) implies (M @) * (i,j) > 0 )

assume [i,j] in Indices (M @) ; :: thesis: (M @) * (i,j) > 0

then A1: [j,i] in Indices M by MATRIX_0:def 6;

then (M @) * (i,j) = M * (j,i) by MATRIX_0:def 6;

hence (M @) * (i,j) > 0 by A1, Def1; :: thesis: verum

end;assume [i,j] in Indices (M @) ; :: thesis: (M @) * (i,j) > 0

then A1: [j,i] in Indices M by MATRIX_0:def 6;

then (M @) * (i,j) = M * (j,i) by MATRIX_0:def 6;

hence (M @) * (i,j) > 0 by A1, Def1; :: thesis: verum