let n be Nat; :: thesis: for K being Field

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

for p being Element of Permutations n ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 0. K )

let K be Field; :: 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

for p being Element of Permutations n ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 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 for p being Element of Permutations n ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 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: for p being Element of Permutations n ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 0. K )

then consider i being Element of NAT such that

A1: i in Seg n and

A2: for k being Element of NAT st k in Seg n holds

(Col (M,i)) . k = 0. K ;

let p be Element of Permutations n; :: thesis: ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 0. K )

n in NAT by ORDINAL1:def 12;

then consider k being Element of NAT such that

A3: k in Seg n and

A4: i = p . k by A1, Th48;

A5: 1 <= k by A3, FINSEQ_1:1;

len M = n by MATRIX_0:def 2;

then k <= len M by A3, FINSEQ_1:1;

then A6: k in dom M by A5, FINSEQ_3:25;

take k ; :: thesis: ( k in Seg n & (Path_matrix (p,M)) . k = 0. K )

len (Path_matrix (p,M)) = n by MATRIX_3:def 7;

then dom (Path_matrix (p,M)) = Seg n by FINSEQ_1:def 3;

then (Path_matrix (p,M)) . k = M * (k,i) by A3, A4, MATRIX_3:def 7;

then (Path_matrix (p,M)) . k = (Col (M,i)) . k by A6, MATRIX_0:def 8;

hence ( k in Seg n & (Path_matrix (p,M)) . k = 0. K ) by A2, A3; :: thesis: verum

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

for p being Element of Permutations n ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 0. K )

let K be Field; :: 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

for p being Element of Permutations n ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 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 for p being Element of Permutations n ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 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: for p being Element of Permutations n ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 0. K )

then consider i being Element of NAT such that

A1: i in Seg n and

A2: for k being Element of NAT st k in Seg n holds

(Col (M,i)) . k = 0. K ;

let p be Element of Permutations n; :: thesis: ex l being Element of NAT st

( l in Seg n & (Path_matrix (p,M)) . l = 0. K )

n in NAT by ORDINAL1:def 12;

then consider k being Element of NAT such that

A3: k in Seg n and

A4: i = p . k by A1, Th48;

A5: 1 <= k by A3, FINSEQ_1:1;

len M = n by MATRIX_0:def 2;

then k <= len M by A3, FINSEQ_1:1;

then A6: k in dom M by A5, FINSEQ_3:25;

take k ; :: thesis: ( k in Seg n & (Path_matrix (p,M)) . k = 0. K )

len (Path_matrix (p,M)) = n by MATRIX_3:def 7;

then dom (Path_matrix (p,M)) = Seg n by FINSEQ_1:def 3;

then (Path_matrix (p,M)) . k = M * (k,i) by A3, A4, MATRIX_3:def 7;

then (Path_matrix (p,M)) . k = (Col (M,i)) . k by A6, MATRIX_0:def 8;

hence ( k in Seg n & (Path_matrix (p,M)) . k = 0. K ) by A2, A3; :: thesis: verum