reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem
  con_class a = con_class b iff con_class a meets con_class b
proof
  thus con_class a = con_class b implies con_class a meets con_class b
  by Th81;
  assume con_class a meets con_class b;
  then consider x being object such that
A1: x in con_class a and
A2: x in con_class b by XBOOLE_0:3;
  reconsider x as Element of G by A1;
A3: a,x are_conjugated by A1,Th81;
  thus con_class a c= con_class b
  proof
    let y be object;
    assume y in con_class a;
    then consider g such that
A4: g = y and
A5: a,g are_conjugated by Th80;
A6: b,x are_conjugated by A2,Th81;
    x,a are_conjugated by A1,Th81;
    then x,g are_conjugated by A5,Th77;
    then b,g are_conjugated by A6,Th77;
    hence thesis by A4,Th80;
  end;
  let y be object;
  assume y in con_class b;
  then consider g such that
A7: g = y and
A8: b,g are_conjugated by Th80;
  x,b are_conjugated by A2,Th81;
  then x,g are_conjugated by A8,Th77;
  then a,g are_conjugated by A3,Th77;
  hence thesis by A7,Th80;
end;
