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;

theorem Th58:
  x in H |^ a iff ex g st x = g |^ a & g in H
proof
  thus x in H |^ a implies ex g st x = g |^ a & g in H
  proof
    assume x in H |^ a;
    then x in the carrier of H |^ a;
    then x in carr H |^ a by Def6;
    then consider b such that
A1: x = b |^ a & b in carr H by Th41;
    take b;
    thus thesis by A1;
  end;
  given g such that
A2: x = g |^ a and
A3: g in H;
  g in carr H by A3;
  then x in carr H |^ a by A2,Th41;
  then x in the carrier of H |^ a by Def6;
  hence thesis;
end;
