
theorem
  for G being Group holds lattice G is complete
proof
  let G be Group;
  let Y be Subset of lattice G;
  per cases;
  suppose
A1: Y = {};
    take Top lattice G;
    thus Top lattice G is_less_than Y
    by A1;
    let b be Element of lattice G;
    assume b is_less_than Y;
    thus thesis by LATTICES:19;
  end;
  suppose
    Y <> {};
    then reconsider X = Y as non empty Subset of Subgroups G;
    reconsider p = meet X as Element of lattice G by GROUP_3:def 1;
    take p;
    set x = the Element of X;
    thus p is_less_than Y
    proof
      let q be Element of lattice G;
      reconsider H = q as strict Subgroup of G by GROUP_3:def 1;
      reconsider h = q as Element of Subgroups G;
      assume
A2:   q in Y;
      carr G.h = the carrier of H by Def1;
      then meet (carr G.:X) c= the carrier of H by A2,FUNCT_2:35,SETFAM_1:3;
      then the carrier of meet X c= the carrier of H by Def2;
      hence thesis by Th25;
    end;
    let r be Element of lattice G;
    reconsider H = r as Subgroup of G by GROUP_3:def 1;
    assume
A3: r is_less_than Y;
A4: for Z1 being set st Z1 in carr G.:X holds the carrier of H c= Z1
    proof
      let Z1 be set;
      assume
A5:   Z1 in carr G.:X;
      then reconsider Z2 = Z1 as Subset of G;
      consider z1 being Element of Subgroups G such that
A6:   z1 in X & Z2 = carr G.z1 by A5,FUNCT_2:65;
      reconsider z1 as Element of lattice G;
      reconsider z3 = z1 as strict Subgroup of G by GROUP_3:def 1;
      Z1 = the carrier of z3 & r [= z1 by A3,A6,Def1;
      hence thesis by Th25;
    end;
    carr G.x in carr G.:X by FUNCT_2:35;
    then the carrier of H c= meet (carr G.:X) by A4,SETFAM_1:5;
    then the carrier of H c= the carrier of meet X by Def2;
    hence thesis by Th25;
  end;
end;
