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
  for a st a in H holds con_class a c= carr H
proof
  let H be strict Subgroup of G;
  thus H is normal Subgroup of G implies for a st a in H holds con_class a c=
  carr H
  proof
    assume
A1: H is normal Subgroup of G;
    let a;
    assume
A2: a in H;
    let x be object;
    assume x in con_class a;
    then consider b such that
A3: x = b and
A4: a,b are_conjugated by Th80;
    consider c such that
A5: b = a |^ c by A4,Th74;
    x in H |^ c by A2,A3,A5,Th58;
    then x in H by A1,Def13;
    hence thesis;
  end;
  assume
A6: for a st a in H holds con_class a c= carr H;
  H is normal
  proof
    let a;
    H |^ a = H
    proof
      let b;
      thus b in H |^ a implies b in H
      proof
        assume b in H |^ a;
        then consider c such that
A7:     b = c |^ a & c in H by Th58;
        b in con_class c & con_class c c= carr H by A6,A7,Th82;
        hence thesis;
      end;
      assume b in H;
      then
A8:   con_class b c= carr H by A6;
      b |^ a" in con_class b by Th82;
      then b |^ a" in H by A8;
      then b |^ a" |^ a in H |^ a by Th58;
      hence thesis by Th25;
    end;
    hence thesis;
  end;
  hence thesis;
end;
