reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,L for Element of K;
reserve V1,V2 for finite-dimensional VectSp of K,
  W1,W2 for Subspace of V1,
  U1 ,U2 for Subspace of V2,
  b1 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,
  bw1 for OrdBasis of W1,
  bw2 for OrdBasis of W2,
  Bu1 for FinSequence of U1,
  Bu2 for FinSequence of U2;

theorem Th21:
  len B1 in dom B1 implies Sum lmlt (Line(Jordan_block(L,len B1),
  len B1),B1)= L*(B1/.(len B1))
proof
  set N=len B1;
  assume
A1: N in dom B1;
  set J=Jordan_block(L,N);
  set ONE=1.(K,N);
  thus Sum lmlt (Line(J,N),B1) = Sum lmlt(L*Line(ONE,N),B1) by Th5
    .= L* Sum lmlt (Line(ONE,N),B1) by MATRLIN2:13
    .= L* (B1/.N) by A1,MATRLIN2:16;
end;
