reserve a,b,c for set;

theorem Th33:
  for X being infinite set, x0 being set for B being Basis of
  DiscrWithInfin(X,x0) holds card X c= card B
proof
  let X be infinite set;
  let x0 be set;
  set T = DiscrWithInfin(X,x0);
  set SX = SmallestPartition X;
A1: card {{x0}} = 1 by CARD_1:30;
A2: card SX = card X by Th12;
  let B be Basis of T;
A3: the carrier of T = X by Def5;
A4: SX = the set of all {x} where x is Element of X by EQREL_1:37;
A5: SX\{{x0}} c= B
  proof
    let a be object;
     reconsider aa=a as set by TARSKI:1;
    assume
A6: a in SX\{{x0}};
    then not a in {{x0}} by XBOOLE_0:def 5;
    then
A7: a <> {x0} by TARSKI:def 1;
    a in SX by A6,XBOOLE_0:def 5;
    then ex x being Element of X st a = {x} by A4;
    then consider x being Element of T such that
A8: a = {x} and
A9: x <> x0 by A3,A7;
A10: x in aa by A8,TARSKI:def 1;
    a is open Subset of T by A3,A8,A9,Th22;
    then consider U being Subset of T such that
A11: U in B and
A12: x in U and
A13: U c= aa by A10,YELLOW_9:31;
    aa c= U by A8,A12,ZFMISC_1:31;
    hence thesis by A11,A13,XBOOLE_0:def 10;
  end;
A14: 1 in card X by CARD_3:86;
  then card X +` 1 = card X by CARD_2:76;
  then card (SX \ {{x0}}) = card X by A1,A2,A14,CARD_2:98;
  hence thesis by A5,CARD_1:11;
end;
