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

(PPath_product M) . 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

(PPath_product M) . 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

(PPath_product M) . 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 (PPath_product M) . 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: (PPath_product M) . 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 A1, FINSEQ_1:def 3;

then A3: Product (Path_matrix (p,M)) = 0. K by A2, FVSUM_1:82;

(PPath_product M) . p = the multF of K $$ (Path_matrix (p,M)) by Def1

.= 0. K by A3, GROUP_4:def 2 ;

hence (PPath_product M) . p = 0. K ; :: thesis: verum

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

(PPath_product M) . 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

(PPath_product M) . 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

(PPath_product M) . 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 (PPath_product M) . 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: (PPath_product M) . 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 A1, FINSEQ_1:def 3;

then A3: Product (Path_matrix (p,M)) = 0. K by A2, FVSUM_1:82;

(PPath_product M) . p = the multF of K $$ (Path_matrix (p,M)) by Def1

.= 0. K by A3, GROUP_4:def 2 ;

hence (PPath_product M) . p = 0. K ; :: thesis: verum