reserve a, b, r, s for Real;
reserve n, m for Nat,
  F for Subset-Family of Closed-Interval-TSpace (r,s);

theorem Th48:
  for L being TopSpace, G, ALL being set st ALL = {C where C is
  Subset-Family of L: C is Cover of L & C c= G} for M being set st M
is_minimal_in RelIncl ALL & M in field RelIncl ALL for A1 being Subset of L st
  A1 in M holds not ex A2, A3 being Subset of L st A2 in M & A3 in M & A1 c= A2
  \/ A3 & A1 <> A2 & A1 <> A3
proof
  let L be TopSpace;
  let G be set;
  let ALL be set such that
A1: ALL = {C where C is Subset-Family of L: C is Cover of L & C c= G};
  set R = RelIncl ALL;
  let M be set such that
A2: M is_minimal_in RelIncl ALL and
A3: M in field RelIncl ALL;
A4: field R = ALL by WELLORD2:def 1;
  then consider C being Subset-Family of L such that
A5: M = C and
A6: C is Cover of L and
A7: C c= G by A1,A3;
  let A1 be Subset of L such that
A8: A1 in M;
  set Y = C \ {A1};
A9: Y <> M by A8,ZFMISC_1:56;
  given A2, A3 being Subset of L such that
A10: A2 in M and
A11: A3 in M and
A12: A1 c= A2 \/ A3 and
A13: A1 <> A2 and
A14: A1 <> A3;
A15: union C = [#]L by A6,SETFAM_1:45;
  union Y = the carrier of L
  proof
    thus union Y c= the carrier of L;
    let x be object;
    assume
A16: x in the carrier of L;
    per cases;
    suppose
A17:  x in A1;
      per cases by A12,A17,XBOOLE_0:def 3;
      suppose
A18:    x in A2;
        A2 in Y by A5,A10,A13,ZFMISC_1:56;
        hence thesis by A18,TARSKI:def 4;
      end;
      suppose
A19:    x in A3;
        A3 in Y by A5,A11,A14,ZFMISC_1:56;
        hence thesis by A19,TARSKI:def 4;
      end;
    end;
    suppose
A20:  not x in A1;
      consider Z being set such that
A21:  x in Z and
A22:  Z in C by A15,A16,TARSKI:def 4;
      Z in Y by A20,A21,A22,ZFMISC_1:56;
      hence thesis by A21,TARSKI:def 4;
    end;
  end;
  then
A23: Y is Cover of L by SETFAM_1:def 11;
A24: Y c= M by A5,XBOOLE_1:36;
  then Y c= G by A5,A7;
  then
A25: Y in ALL by A1,A23;
  then [Y,M] in R by A4,A3,A24,WELLORD2:def 1;
  hence contradiction by A4,A2,A9,A25;
end;
