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 Th134:
  for H being strict Subgroup of G holds x in Normalizer H iff
  ex h st x = h & H |^ h = H
proof
  let H be strict Subgroup of G;
  thus x in Normalizer H implies ex h st x = h & H |^ h = H
  proof
    assume x in Normalizer H;
    then consider a such that
A1: x = a and
A2: carr H |^ a = carr H by Th129;
    H |^ a = H by A2,Def6;
    hence thesis by A1;
  end;
  given h such that
A3: x = h and
A4: H |^ h = H;
  carr H |^ h = carr H by A4,Def6;
  hence thesis by A3,Th129;
end;
