reserve A,B for limit_ordinal infinite Ordinal;
reserve B1,B2,B3,B5,B6,D, C for Ordinal;
reserve X for set;
reserve X for Subset of A;

theorem Th9:
  X is unbounded & B1 in A implies LBound(B1,X) in X & B1 in LBound(B1,X)
proof
  assume
A1: X is unbounded;
  defpred P[set] means $1 in X & B1 in $1;
  set LB = { B2 where B2 is Element of A: P[B2]};
A2: for x being set st x in LB holds B1 in x
  proof
    let x be set;
    assume
A3: x in LB;
    P[x] from CARD_FIL:sch 1(A3);
    hence thesis;
  end;
  LB is Subset of A from DOMAIN_1:sch 7;
  then
A4: inf LB = meet On LB & On LB = LB by ORDINAL2:def 2,ORDINAL3:6;
  assume
A5: B1 in A;
  X is non empty by A1,Th7;
  hence LBound(B1,X) in X;
  ex B3 being Element of A st B3 in LB by A1,A5,Th8;
  then B1 in meet LB by A2,SETFAM_1:def 1;
  hence thesis by A1,A5,A4,Def6;
end;
