
theorem Th34:
  for S be non empty finite set,
  X be Subset of S,
  s,t be FinSequence of S,
  SD be Subset of dom s,
  x be Subset of X
  st SD = s"X &t = extract(s,SD) holds
  card (s"x) = card (t"x)
  proof
    let S be non empty finite set,
    X be Subset of S,
    s,t be FinSequence of S,
    SD be Subset of dom s,
    x be Subset of X;
    assume A1: SD = s"X &t = extract(s,SD);
    reconsider SD as finite set;
    set f= (canFS SD)";
    len t = card SD by A1,Th11;then
    A2: dom t = Seg (card SD) by FINSEQ_1:def 3;
    then
    reconsider g= f as Function of SD, dom t by FINSEQ_1:95;
    A3: (canFS SD).:(t"x) = f"(t"x) by FUNCT_1:84
    .= (t*f)"x by RELAT_1:146
    .= (s|SD)"x by A1,Th12
    .= SD /\ s"x by FUNCT_1:70
    .= s"x by A1,RELAT_1:143,XBOOLE_1:28; then
    A4: card (s"x) c= card (t"x) by CARD_1:66;
    A5: t"x c= Seg (card SD) by A2,RELAT_1:132;
    len canFS(SD) = card SD by FINSEQ_1:93;then
    A6: t"x is Subset of dom (canFS SD) by A5,FINSEQ_1:def 3;
    A7: (f*(canFS SD)) = id dom (canFS SD) by FUNCT_1:39;
    f.:(s"x) =(f*(canFS SD)).:(t"x) by A3,RELAT_1:126
    .= t"x by A6,A7,FUNCT_1:92; then
    card (t"x) c= card (s"x) by CARD_1:66;
    hence thesis by A4,XBOOLE_0:def 10;
  end;
