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 Th129:
  x in Normalizer A iff ex h st x = h & A |^ h = A
proof
  thus x in Normalizer A implies ex h st x = h & A |^ h = A
  proof
    assume x in Normalizer A;
    then x in the carrier of Normalizer A;
    then x in {h : A |^ h = A} by Def14;
    hence thesis;
  end;
  given h such that
A1: x = h & A |^ h = A;
  x in {b : A |^ b = A} by A1;
  then x in the carrier of Normalizer A by Def14;
  hence thesis;
end;
