let n be Nat; :: thesis: for K being Field
for a being Element of K
for M, N 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 = a * ((Col (N,i)) /. k) ) & ( for l being Element of NAT st l <> i & l in Seg n holds
Col (M,l) = Col (N,l) ) ) holds
for p being Element of Permutations n ex l being Element of NAT st
( l in Seg n & (Path_matrix (p,M)) /. l = a * ((Path_matrix (p,N)) /. l) )

let K be Field; :: thesis: for a being Element of K
for M, N 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 = a * ((Col (N,i)) /. k) ) & ( for l being Element of NAT st l <> i & l in Seg n holds
Col (M,l) = Col (N,l) ) ) holds
for p being Element of Permutations n ex l being Element of NAT st
( l in Seg n & (Path_matrix (p,M)) /. l = a * ((Path_matrix (p,N)) /. l) )

let a be Element of K; :: thesis: for M, N 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 = a * ((Col (N,i)) /. k) ) & ( for l being Element of NAT st l <> i & l in Seg n holds
Col (M,l) = Col (N,l) ) ) holds
for p being Element of Permutations n ex l being Element of NAT st
( l in Seg n & (Path_matrix (p,M)) /. l = a * ((Path_matrix (p,N)) /. l) )

let M, N 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 = a * ((Col (N,i)) /. k) ) & ( for l being Element of NAT st l <> i & l in Seg n holds
Col (M,l) = Col (N,l) ) ) implies for p being Element of Permutations n ex l being Element of NAT st
( l in Seg n & (Path_matrix (p,M)) /. l = a * ((Path_matrix (p,N)) /. l) ) )

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 = a * ((Col (N,i)) /. k) ) & ( for l being Element of NAT st l <> i & l in Seg n holds
Col (M,l) = Col (N,l) ) ) ; :: thesis: for p being Element of Permutations n ex l being Element of NAT st
( l in Seg n & (Path_matrix (p,M)) /. l = a * ((Path_matrix (p,N)) /. l) )

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 = a * ((Col (N,i)) /. k) and
for l being Element of NAT st l <> i & l in Seg n holds
Col (M,l) = Col (N,l) ;
let p be Element of Permutations n; :: thesis: ex l being Element of NAT st
( l in Seg n & (Path_matrix (p,M)) /. l = a * ((Path_matrix (p,N)) /. l) )

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 ;
A5: 1 <= k by ;
len (Path_matrix (p,N)) = n by MATRIX_3:def 7;
then A6: k in dom (Path_matrix (p,N)) by ;
then (Path_matrix (p,N)) . k = N * (k,i) by ;
then A7: (Path_matrix (p,N)) /. k = N * (k,i) by ;
len (Col (N,i)) = len N by MATRIX_0:def 8;
then A8: dom (Col (N,i)) = dom N by FINSEQ_3:29;
len N = n by MATRIX_0:def 2;
then k <= len N by ;
then A9: k in dom N by ;
then (Col (N,i)) . k = N * (k,i) by MATRIX_0:def 8;
then A10: (Col (N,i)) /. k = (Path_matrix (p,N)) /. k by ;
len M = n by MATRIX_0:def 2;
then k <= len M by ;
then A11: k in dom M by ;
take k ; :: thesis: ( k in Seg n & (Path_matrix (p,M)) /. k = a * ((Path_matrix (p,N)) /. k) )
len (Path_matrix (p,M)) = n by MATRIX_3:def 7;
then A12: dom (Path_matrix (p,M)) = Seg n by FINSEQ_1:def 3;
then (Path_matrix (p,M)) . k = M * (k,i) by ;
then (Path_matrix (p,M)) . k = (Col (M,i)) . k by
.= a * ((Path_matrix (p,N)) /. k) by A2, A3, A10 ;
hence ( k in Seg n & (Path_matrix (p,M)) /. k = a * ((Path_matrix (p,N)) /. k) ) by ; :: thesis: verum