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;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;

theorem
  for H being strict Subgroup of G holds H is normal Subgroup of G iff
  con_class H = {H}
proof
  let H be strict Subgroup of G;
  thus H is normal Subgroup of G implies con_class H = {H}
  proof
    assume
A1: H is normal Subgroup of G;
    thus con_class H c= {H}
    proof
      let x be object;
      assume x in con_class H;
      then consider H1 being strict Subgroup of G such that
A2:   x = H1 and
A3:   H,H1 are_conjugated by Def12;
      ex g st H1 = H |^ g by A3,Th102;
      then x = H by A1,A2,Def13;
      hence thesis by TARSKI:def 1;
    end;
    H in con_class H by Th109;
    hence thesis by ZFMISC_1:31;
  end;
  assume
A4: con_class H = {H};
  H is normal
  proof
    let a;
    H |^ a in con_class H by Th108;
    hence thesis by A4,TARSKI:def 1;
  end;
  hence thesis;
end;
