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
  proof
A1: A in con_class A;
    assume con_class A = con_class B;
    hence thesis by A1;
  end;
  assume con_class A meets con_class B;
  then consider x being object such that
A2: x in con_class A and
A3: x in con_class B by XBOOLE_0:3;
  reconsider x as Subset of G by A2;
A4: A,x are_conjugated by A2,Th95;
  thus con_class A c= con_class B
  proof
    let y be object;
    assume y in con_class A;
    then consider C such that
A5: C = y and
A6: A,C are_conjugated;
A7: B,x are_conjugated by A3,Th95;
    x,A are_conjugated by A2,Th95;
    then x,C are_conjugated by A6,Th91;
    then B,C are_conjugated by A7,Th91;
    hence thesis by A5;
  end;
  let y be object;
  assume y in con_class B;
  then consider C such that
A8: C = y and
A9: B,C are_conjugated;
  x,B are_conjugated by A3,Th95;
  then x,C are_conjugated by A9,Th91;
  then A,C are_conjugated by A4,Th91;
  hence thesis by A8;
end;
