reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,L for Element of K;

theorem Th11:
  <*Jordan_block(L,n)*> is FinSequence_of_Jordan_block of L,K
proof
  now
A1: dom <*Jordan_block(L,n)*> ={1} by FINSEQ_1:2,def 8;
    let i;
    assume i in dom <*Jordan_block(L,n)*>;
    then <*Jordan_block(L,n)*>.1=Jordan_block(L,n) & i=1 by A1,TARSKI:def 1;
    hence ex a,k st <*Jordan_block(L,n)*>.i=Jordan_block(a,k);
  end;
  then reconsider
  JJ=<*Jordan_block(L,n)*> as FinSequence_of_Jordan_block of K by Def2;
  now
A2: dom JJ ={1} by FINSEQ_1:2,def 8;
    let i;
    assume i in dom JJ;
    then JJ.1=Jordan_block(L,n) & i=1 by A2,TARSKI:def 1;
    hence ex n st JJ.i = Jordan_block(L,n);
  end;
  hence thesis by Def3;
end;
