reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem Th34:
  the carrier of gr A = meet{B : ex H being strict Subgroup of G
  st B = the carrier of H & A c= carr H}
proof
  defpred P[Subgroup of G] means A c= carr $1;
  set X = {B :ex H being strict Subgroup of G st B = the carrier of H & A c=
  carr H};
A1: now
    let Y;
    assume Y in X;
    then
    ex B st Y = B & ex H being strict Subgroup of G st B = the carrier of
    H & A c= carr H;
    hence A c= Y;
  end;
  the carrier of (Omega).G = carr (Omega).G;
  then
A2: ex H being strict Subgroup of G st P[H];
  consider H being strict Subgroup of G such that
A3: the carrier of H = meet{B : ex H being strict Subgroup of G st B =
  the carrier of H & P[H]} from MeetSbgEx(A2);
A4: now
    let H1 be strict Subgroup of G;
A5: the carrier of H1 = carr H1;
    assume A c= the carrier of H1;
    then the carrier of H1 in X by A5;
    hence H is Subgroup of H1 by A3,GROUP_2:57,SETFAM_1:3;
  end;
  carr (Omega).G in X;
  then A c= the carrier of H by A3,A1,SETFAM_1:5;
  hence thesis by A3,A4,Def4;
end;
