reserve F for Field;
reserve VS for strict VectSp of F;
reserve u,e for set;
reserve x for set;
reserve Z1 for set;

theorem
  for VS holds lattice VS is complete
proof
  let VS;
  let Y be Subset of lattice VS;
  per cases;
  suppose
A1: Y = {};
    thus thesis
    proof
      take Top lattice VS;
      thus Top lattice VS is_less_than Y
      by A1;
      let b be Element of lattice VS;
      assume b is_less_than Y;
      thus thesis by LATTICES:19;
    end;
  end;
  suppose
    Y <> {};
    then reconsider X = Y as non empty Subset of Subspaces(VS);
    reconsider p = meet X as Element of lattice VS by VECTSP_5:def 3;
    take p;
    set x = the Element of X;
    thus p is_less_than Y
    proof
      let q be Element of lattice VS;
      consider H being strict Subspace of VS such that
A2:   H=q by VECTSP_5:def 3;
      reconsider h = q as Element of Subspaces(VS);
      assume
A3:   q in Y;
      (carr VS).h = the carrier of H by A2,Def4;
      then meet ((carr VS).:X) c= the carrier of H by A3,FUNCT_2:35,SETFAM_1:3;
      then the carrier of meet X c= the carrier of H by Def5;
      hence p [= q by A2,Th6;
    end;
    let r be Element of lattice VS;
    consider H being strict Subspace of VS such that
A4: H=r by VECTSP_5:def 3;
    assume
A5: r is_less_than Y;
A6: for Z1 st Z1 in (carr VS).:X holds the carrier of H c= Z1
    proof
      let Z1;
      assume
A7:   Z1 in (carr VS).:X;
      then reconsider Z2=Z1 as Subset of VS;
      consider z1 being Element of Subspaces(VS) such that
A8:   z1 in X and
A9:   Z2=(carr VS).z1 by A7,FUNCT_2:65;
      reconsider z1 as Element of lattice VS;
A10:  r [= z1 by A5,A8;
      consider z3 being strict Subspace of VS such that
A11:  z3=z1 by VECTSP_5:def 3;
      Z1 = the carrier of z3 by A9,A11,Def4;
      hence thesis by A4,A11,A10,Th6;
    end;
    (carr VS).x in (carr VS).:X by FUNCT_2:35;
    then the carrier of H c= meet ((carr VS).:X) by A6,SETFAM_1:5;
    then the carrier of H c= the carrier of meet X by Def5;
    hence r [= p by A4,Th6;
  end;
end;
