
theorem Th34:
  for S being non empty doubleLoopStr, F being non empty Subset of
  S, lc being non empty RightLinearCombination of F ex p being
RightLinearCombination of F, e being Element of S st lc = p^<* e *> & <*e*> is
  RightLinearCombination of F
proof
  let S be non empty doubleLoopStr, F be non empty Subset of S, lc be non
  empty RightLinearCombination of F;
  len lc <> 0;
  then consider
  p being FinSequence of the carrier of S, e being Element of S such
  that
A1: lc = p^<*e*> by FINSEQ_2:19;
  now
    let i be set;
    assume
A2: i in dom p;
    then reconsider i1=i as Element of NAT;
A3: dom p c= dom lc by A1,FINSEQ_1:26;
    then consider u being Element of S, a being Element of F such that
A4: lc/.i = a*u by A2,Def10;
    take u, a;
    thus p/.i = p.i by A2,PARTFUN1:def 6
      .= lc.i1 by A1,A2,FINSEQ_1:def 7
      .= a*u by A2,A3,A4,PARTFUN1:def 6;
  end;
  then reconsider p as RightLinearCombination of F by Def10;
A5: len lc = len p +1 by A1,FINSEQ_2:16;
  take p;
  take e;
  thus lc = p^<* e *> by A1;
  let i be set such that
A6: i in dom <*e*>;
A7: len lc in dom lc by FINSEQ_5:6;
  then
A8: lc/.(len lc) = lc.(len lc) by PARTFUN1:def 6;
  dom <*e*> = {1} by FINSEQ_1:2,38;
  then
A9: i = 1 by A6,TARSKI:def 1;
  consider u being Element of S, a being Element of F such that
A10: lc/.(len lc) = a*u by A7,Def10;
  take u, a;
  thus <*e*>/.i = <*e*>.i by A6,PARTFUN1:def 6
    .= e by A9
    .= a*u by A1,A5,A10,A8,FINSEQ_1:42;
end;
