reserve n,k,b for Nat, i for Integer;

theorem Th9:
  for F being XFinSequence
  for X,Y being set st X misses Y
  ex P being Permutation of dom SubXFinS(F,X\/Y) st
  SubXFinS(F,X\/Y) * P = SubXFinS(F,X) ^ SubXFinS(F,Y)
  proof
    let F be XFinSequence;
    let X,Y be set such that A1: X misses Y;
    set A=Sgm0(X/\Segm len F)^Sgm0(Y/\Segm len F);
    set B=Sgm0((X\/Y)/\Segm len F);
    A2: rng Sgm0(X/\Segm len F) = X/\len F &
    rng Sgm0(Y/\Segm len F) = Y/\len F by AFINSQ_2:def 4;
    then reconsider A as one-to-one Function by A1,XBOOLE_1:76,CARD_FIN:52;
    reconsider B as one-to-one Function;
    A3: (X\/Y) /\ len F = (X/\len F) \/ (Y/\len F) by XBOOLE_1:23;
    then A4: rng A = rng B by Th5;
    then A5: rng A = dom (B") by FUNCT_1:33;
    set P=(B")*A;
    A6: dom P = dom A by A5,RELAT_1:27
    .= dom B by A3,A1,XBOOLE_1:76,Th4
    .= dom SubXFinS(F,X\/Y) by Th6;
    A7: rng P = rng (B") by A5,RELAT_1:28
    .= dom B by FUNCT_1:33
    .= dom SubXFinS(F,X\/Y) by Th6;
    reconsider P as Function of dom SubXFinS(F,X\/Y),dom SubXFinS(F,X\/Y)
    by A6,A7,FUNCT_2:2;
    card dom SubXFinS(F,X\/Y) = card dom SubXFinS(F,X\/Y);
    then P is onto by FINSEQ_4:63;
    then reconsider P as Permutation of dom SubXFinS(F,X\/Y);
    take P;
    A8: rng Sgm0(X/\Segm len F) c= len F & rng Sgm0(Y/\Segm len F) c= len F
    by A2,XBOOLE_1:17;
    thus SubXFinS(F,X\/Y) * P = F * (Sgm0((X\/Y)/\Segm len F) * (B"*A))
    by RELAT_1:36
    .= F * ((Sgm0((X\/Y)/\Segm len F) * B") * A) by RELAT_1:36
    .= F * (id (rng A) * A) by FUNCT_1:39,A4
    .= F * A by RELAT_1:54
    .= SubXFinS(F,X) ^ SubXFinS(F,Y) by A8,AFINSQ_2:70;
  end;
