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
  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;
    hence thesis by A4,A6,Th77,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;
  hence thesis by A3,A7,Th77,Th80;
end;
