theorem
  con_class a = {0_G} iff a = 0_G
proof
  thus con_class a = {0_G} implies a = 0_G
  proof
    assume
A1: con_class a = {0_G};
    a in con_class a by Th81;
    hence thesis by A1,TARSKI:def 1;
  end;
  assume
A2: a = 0_G;
  thus con_class a c= {0_G}
  proof
    let x be object;
    assume x in con_class a;
    then consider b such that
A3: b = x and
A4: a,b are_conjugated by Th80;
    b = 0_G by A2,A4,ThB78;
    hence thesis by A3,TARSKI:def 1;
  end;
  thus thesis by A2,Th81,ZFMISC_1:31;
end;
