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
  A <> {} implies (1_G) |^ A = {1_G}
proof
  set y = the Element of A;
  assume
A1: A <> {};
  then reconsider y as Element of G by TARSKI:def 3;
  thus (1_G) |^ A c= {1_G}
  proof
    let x be object;
    assume x in (1_G) |^ A;
    then ex a st x = (1_G) |^ a & a in A by Th42;
    then x = 1_G by Th17;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {1_G};
  then x = 1_G by TARSKI:def 1;
  then (1_G) |^ y = x by Th17;
  hence thesis by A1,Th42;
end;
