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

theorem Th12:
  for J be FinSequence_of_Jordan_block of L,K holds J (+) mlt(len
  J|->a,1.(K,Len J)) is FinSequence_of_Jordan_block of (L+a),K
proof
  let J be FinSequence_of_Jordan_block of L,K;
  set M=mlt(len J|->a,1.(K,Len J));
A1: for i st i in dom (J(+)M) ex n st (J(+)M).i=Jordan_block(L+a,n)
  proof
A2: dom M=dom 1.(K,Len J) by MATRIXJ1:def 9;
A3: dom J=Seg len J by FINSEQ_1:def 3;
A4: dom 1.(K,Len J)=dom Len J by MATRIXJ1:def 8;
    let i such that
A5: i in dom (J(+)M);
A6: i in dom J by A5,MATRIXJ1:def 10;
    then consider n such that
A7: J.i = Jordan_block(L,n) by Def3;
    take n;
A8: len (J.i)=n by A7,MATRIX_0:24;
A9: dom Len J=dom J by MATRIXJ1:def 3;
    then
A10: (Len J).i=len (J.i) by A6,MATRIXJ1:def 3;
    len (len J|->a)=len J by CARD_1:def 7;
    then dom (len J|->a)=dom J by FINSEQ_3:29;
    then (len J|->a)/.i = (len J|->a).i by A6,PARTFUN1:def 6
      .= a by A6,A3,FINSEQ_2:57;
    then M.i = a * 1.(K,Len J).i by A6,A2,A4,A9,MATRIXJ1:def 9
      .= a* 1.(K,n) by A6,A4,A9,A10,A8,MATRIXJ1:def 8;
    hence (J(+)M).i = Jordan_block(L,n) +a* 1.(K,n) by A5,A7,MATRIXJ1:def 10
      .= Jordan_block(L+a,n) by Th9;
  end;
  J(+)M is Jordan-block-yielding
  proof
    let i;
    assume i in dom (J(+)M);
    then ex n st (J(+)M).i=Jordan_block(L+a,n) by A1;
    hence thesis;
  end;
  then reconsider JM=J(+)M as FinSequence_of_Jordan_block of K;
  JM is FinSequence_of_Jordan_block of (L+a),K
  proof
    let i;
    assume i in dom JM;
    hence thesis by A1;
  end;
  hence thesis;
end;
