reserve A for set, x,y,z for object,
  k for Element of NAT;

theorem Th10:
  for X being set, A being finite Subset of X, R being Order of X
  st R linearly_orders A holds len(SgmX(R, A)) = card A
proof
  let X be set, A be finite Subset of X, R be Order of X;
  set f = SgmX(R, A);
A1: dom f = Seg(len f) by FINSEQ_1:def 3;
A2: card Seg(len f) = card (len f) by CARD_1:5,FINSEQ_1:54;
  assume
A3: R linearly_orders A;
  then
A4: f is one-to-one by Th9;
  rng f = A by A3,Def2;
  hence thesis by A2,A1,A4,CARD_1:5,WELLORD2:def 4;
end;
