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 = {1_G} iff a = 1_G
proof
  thus con_class a = {1_G} implies a = 1_G
  proof
    assume
A1: con_class a = {1_G};
    a in con_class a by Th81;
    hence thesis by A1,TARSKI:def 1;
  end;
  assume
A2: a = 1_G;
  thus con_class a c= {1_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 = 1_G by A2,A4,Th78;
    hence thesis by A3,TARSKI:def 1;
  end;
  1_G in con_class a by A2,Th81;
  hence thesis by ZFMISC_1:31;
end;
