reserve x for set,
  i,j,k,n for Nat,
  K for Field;
reserve a,b,c,d for Element of K;
reserve D for non empty set;

theorem Th49:
  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 holds ex l being Element of NAT st l in
  Seg n & Path_matrix (p,M).l = 0.K
proof
  let M be Matrix of n, 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;
  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;
  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;
  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 thesis by A2,A3;
end;
