let n be Nat; :: thesis: for K being Field
for p being Element of Permutations n
for M being Matrix of n,K st ex i being Element of NAT st
( i in Seg n & ( for k being Element of NAT st k in Seg n holds
(Col (M,i)) . k = 0. K ) ) holds
() . p = 0. K

let K be Field; :: thesis: for p being Element of Permutations n
for M being Matrix of n,K st ex i being Element of NAT st
( i in Seg n & ( for k being Element of NAT st k in Seg n holds
(Col (M,i)) . k = 0. K ) ) holds
() . p = 0. K

let p be Element of Permutations n; :: thesis: for M being Matrix of n,K st ex i being Element of NAT st
( i in Seg n & ( for k being Element of NAT st k in Seg n holds
(Col (M,i)) . k = 0. K ) ) holds
() . p = 0. K

let M be Matrix of n,K; :: thesis: ( ex i being Element of NAT st
( i in Seg n & ( for k being Element of NAT st k in Seg n holds
(Col (M,i)) . k = 0. K ) ) implies () . p = 0. K )

assume ex i being Element of NAT st
( i in Seg n & ( for k being Element of NAT st k in Seg n holds
(Col (M,i)) . k = 0. K ) ) ; :: thesis: () . p = 0. K
then consider l being Element of NAT such that
A1: l in Seg n and
A2: (Path_matrix (p,M)) . l = 0. K by Th49;
len (Path_matrix (p,M)) = n by MATRIX_3:def 7;
then l in dom (Path_matrix (p,M)) by ;
then A3: Product (Path_matrix (p,M)) = 0. K by ;
() . p = the multF of K \$\$ (Path_matrix (p,M)) by Def1
.= 0. K by ;
hence (PPath_product M) . p = 0. K ; :: thesis: verum