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
  for M,N being Matrix of n, K st i in Seg n for p being Element of
Permutations n holds ex k being Element of NAT st k in Seg n & i = p.k & Col(N,
  i)/.k = Path_matrix (p, N)/.k
proof
  let M,N be Matrix of n, K;
  assume
A1: i in Seg n;
  let p be Element of Permutations n;
  n in NAT by ORDINAL1:def 12;
  then consider k being Element of NAT such that
A2: k in Seg n and
A3: i = p.k by A1,Th48;
  len Path_matrix (p, N) = n by MATRIX_3:def 7;
  then
A4: k in dom Path_matrix (p, N) by A2,FINSEQ_1:def 3;
  then Path_matrix (p, N).k = N*(k,i) by A3,MATRIX_3:def 7;
  then
A5: Path_matrix (p, N)/.k = N*(k,i) by A4,PARTFUN1:def 6;
  take k;
A6: len N = n by MATRIX_0:def 2;
  then k in dom N by A2,FINSEQ_1:def 3;
  then
A7: Col(N,i).k = N*(k,i) by MATRIX_0:def 8;
  len Col(N,i) = len N by MATRIX_0:def 8;
  then k in dom Col(N,i) by A2,A6,FINSEQ_1:def 3;
  hence thesis by A2,A3,A5,A7,PARTFUN1:def 6;
end;
