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 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))) for p being Element of Permutations n holds
  ex l being Element of NAT st l in Seg n & Path_matrix (p,M)/.l = a * (
  Path_matrix (p,N)/.l)
proof
  let a be Element of K;
  let M,N 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 = 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);
  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;
  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 Path_matrix (p, N) = n by MATRIX_3:def 7;
  then
A6: k in dom Path_matrix (p, N) by A3,FINSEQ_1:def 3;
  then Path_matrix (p, N).k = N*(k,i) by A4,MATRIX_3:def 7;
  then
A7: Path_matrix (p, N)/.k = N*(k,i) by A6,PARTFUN1:def 6;
  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 A3,FINSEQ_1:1;
  then
A9: k in dom N by A5,FINSEQ_3:25;
  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 A9,A7,A8,PARTFUN1:def 6;
  len M = n by MATRIX_0:def 2;
  then k <= len M by A3,FINSEQ_1:1;
  then
A11: k in dom M by A5,FINSEQ_3:25;
  take 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 A3,A4,MATRIX_3:def 7;
  then Path_matrix (p, M).k = Col(M,i).k by A11,MATRIX_0:def 8
    .= a * (Path_matrix (p, N)/.k) by A2,A3,A10;
  hence thesis by A12,A3,PARTFUN1:def 6;
end;
