reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for addGroup-like non empty addMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for addGroup;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve x,y,y1,y2 for set;
reserve G for addGroup;
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;
