reserve a,b,c for set;

theorem Th32:
  for X being infinite set, x0 being set for B0 being Basis of
DiscrWithInfin(X,x0) st B0 = ((SmallestPartition X) \ {{x0}}) \/ {F` where F is
  Subset of X: F is finite} holds card B0 = card X
proof
  let X be infinite set;
  let x0 be set;
  set T = DiscrWithInfin(X,x0);
  let B0 be Basis of T;
  set SX = SmallestPartition X, FX = {F` where F is Subset of X: F is finite};
  assume
A1: B0 = (SX \ {{x0}}) \/ FX;
A2: card SX = card X by Th12;
A3: card {{x0}} = 1 by CARD_1:30;
A4: 1 in card X by CARD_3:86;
  then card X +` 1 = card X by CARD_2:76;
  then
A5: card (SX \ {{x0}}) = card X by A3,A2,A4,CARD_2:98;
  card FX = card X by Th31;
  then card B0 c= card X +` card X by A1,A5,CARD_2:34;
  hence card B0 c= card X by CARD_2:75;
  thus thesis by A1,A5,CARD_1:11,XBOOLE_1:7;
end;
