
theorem
  for R, S being non empty multLoopStr, F being non empty Subset of R,
  lc being LinearCombination of F, G being non empty Subset of S, P being
  Function of the carrier of R, the carrier of S, E being FinSequence of [:the
carrier of R, the carrier of R, the carrier of R:] st P.:F c= G & E represents
  lc holds ex LC being LinearCombination of G st len lc = len LC & for i being
  set st i in dom LC
   holds LC.i = (P.((E/.i)`1_3))*(P.((E/.i)`2_3))*(P.((E/.i)`3_3))
proof
  let R, S be non empty multLoopStr, F be non empty Subset of R, lc be
  LinearCombination of F, G be non empty Subset of S, P be Function of the
carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the
  carrier of R, the carrier of R:];
  assume
A1: P.:F c= G;
  deffunc F(Nat)=(P.((E/.$1)`1_3))*(P.((E/.$1)`2_3))*(P.((E/.$1)`3_3));
  consider LC being FinSequence of the carrier of S such that
A2: len LC = len lc and
A3: for k being Nat st k in dom LC holds LC.k = F(k) from FINSEQ_2:sch 1;
  assume
A4: E represents lc;
  now
    let i be set such that
A5: i in dom LC;
    dom lc = dom LC by A2,FINSEQ_3:29;
    then dom P = the carrier of R & (E/.i)`2_3 in F
     by A4,A5,FUNCT_2:def 1;
    then P.((E/.i)`2_3) in P.:F by FUNCT_1:def 6;
    then reconsider a = P.((E/.i)`2_3) as Element of G by A1;
    reconsider u = P.((E/.i)`1_3), v = P.((E/.i)`3_3) as Element of S;
    take u, v, a;
    LC.i = LC/.i by A5,PARTFUN1:def 6;
    hence LC/.i = u*a*v by A3,A5;
  end;
  then reconsider LC as LinearCombination of G by Def8;
  take LC;
  thus len lc = len LC by A2;
  let i be set;
  assume i in dom LC;
  hence thesis by A3;
end;
