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;
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;
    hence thesis by A1,Def13;
  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,Th80,Th74;
        hence thesis;
      end;
      assume b in H; then
A8:   con_class b c= carr H by A6;
      b * (-a) in con_class b by Th80,Th74;
      then b * (-a) in H by A8;
      then b * (-a) * a in H * a by Th58;
      hence thesis by ThB25;
    end;
    hence thesis;
  end;
  hence thesis;
end;
