reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;
reserve F,F1,F2 for FinSequence_of_Matrix of D,
  G,G9,G1,G2 for FinSequence_of_Matrix of K;
reserve S,S1,S2 for FinSequence_of_Square-Matrix of D,
  R,R1,R2 for FinSequence_of_Square-Matrix of K;
reserve N for (Matrix of n,K),
  N1 for (Matrix of m,K);

theorem Th57:
  for i be Element of NAT holds 1.(K,<*i*>) = <*1.(K,i)*>
proof
  let i be Element of NAT;
A1: dom (1.(K,<*i*>)) = dom <*i*> by Def8
    .= Seg 1 by FINSEQ_1:38;
  then 1 in dom (1.(K,<*i*>));
  then
A2: (1.(K,<*i*>)).1 = 1.(K,<*i*>.1) by Def8
    .= 1.(K,i);
  len (1.(K,<*i*>))=1 by A1,FINSEQ_1:def 3;
  hence thesis by A2,FINSEQ_1:40;
end;
