
theorem Th20:
  for L being non empty multMagma for B being non empty AlgebraStr
over L for A being non empty Subset of B for X being Subset-Family of B st (for
Y be set holds Y in X iff Y c= the carrier of B & ex C being Subalgebra of B st
  Y = the carrier of C & A c= Y) holds meet X is opers_closed
proof
  let L being non empty multMagma;
  let B being non empty AlgebraStr over L;
  let A being non empty Subset of B;
  let X being Subset-Family of B such that
A1: for Y be set holds Y in X iff Y c= the carrier of B & ex C being
  Subalgebra of B st Y = the carrier of C & A c= Y;
  B is Subalgebra of B by Th11;
  then
A2: X <> {} by A1;
A3: for x,y being Element of B st x in meet X & y in meet X holds x + y in
  meet X
  proof
    let x,y be Element of B such that
A4: x in meet X & y in meet X;
    now
      reconsider x9 = x, y9 = y as Element of B;
      let Y be set;
      assume
A5:   Y in X;
      then consider C be Subalgebra of B such that
A6:   Y = the carrier of C and
A7:   A c= Y by A1;
      reconsider C as non empty Subalgebra of B by A6,A7;
      reconsider x1 = x9, y1 = y9 as Element of C by A4,A5,A6,SETFAM_1:def 1;
      x + y = x1+ y1 by Th15;
      hence x + y in Y by A6;

    end;
    hence thesis by A2,SETFAM_1:def 1;
  end;
  for a being Element of L, v being Element of B st v in meet X holds a *
  v in meet X
  proof
    let a be Element of L, v be Element of B such that
A8: v in meet X;
    now
      let Y be set;
      assume
A9:   Y in X;
      then consider C be Subalgebra of B such that
A10:  Y = the carrier of C and
A11:  A c= Y by A1;
      reconsider C as non empty Subalgebra of B by A10,A11;
      reconsider v9 = v as Element of C by A8,A9,A10,SETFAM_1:def 1;
      a * v = a * v9 by Th17;
      hence a * v in Y by A10;

    end;
    hence thesis by A2,SETFAM_1:def 1;
  end;
  hence meet X is linearly-closed by A3,VECTSP_4:def 1;
  thus for x,y being Element of B st x in meet X & y in meet X holds x*y in
  meet X
  proof
    let x,y be Element of B such that
A12: x in meet X & y in meet X;
    now
      reconsider x9 = x, y9 = y as Element of B;
      let Y be set;
      assume
A13:  Y in X;
      then consider C be Subalgebra of B such that
A14:  Y = the carrier of C and
A15:  A c= Y by A1;
      reconsider C as non empty Subalgebra of B by A14,A15;
      reconsider x1 = x9, y1 = y9 as Element of C by A12,A13,A14,SETFAM_1:def 1
;
      x*y = x1* y1 by Th16;
      hence x*y in Y by A14;

    end;
    hence thesis by A2,SETFAM_1:def 1;
  end;
  now
    let Y be set;
    assume Y in X;
    then consider C be Subalgebra of B such that
A16: Y = the carrier of C and
A17: A c= Y by A1;
    reconsider C as non empty Subalgebra of B by A16,A17;
    1.B = 1.C by Def3;
    hence 1.B in Y by A16;

  end;
  hence 1.B in meet X by A2,SETFAM_1:def 1;
  now
    let Y be set;
    assume Y in X;
    then consider C be Subalgebra of B such that
A18: Y = the carrier of C and
A19: A c= Y by A1;
    reconsider C as non empty Subalgebra of B by A18,A19;
    0.B = 0.C by Def3;
    hence 0.B in Y by A18;

  end;
  hence thesis by A2,SETFAM_1:def 1;
end;
