reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem Th43:
  for N being strict normal Subgroup of G holds Ker nat_hom N = N
proof
  let N be strict normal Subgroup of G;
  let a;
  thus a in Ker nat_hom N implies a in N
  proof
    assume a in Ker nat_hom N;
    then
A1: (nat_hom N).a = 1_(G./.N) by Th41;
    (nat_hom N).a = a * N & 1_(G./.N) = carr N by Def8,Th24;
    hence thesis by A1,GROUP_2:113;
  end;
  assume a in N;
  then
A2: a * N = carr N by GROUP_2:113;
  (nat_hom N).a = a * N & 1_(G./.N) = carr N by Def8,Th24;
  hence thesis by A2,Th41;
end;
