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 Th41:
  x in A |^ g iff ex h st x = h |^ g & h in A
proof
  thus x in A |^ g implies ex h st x = h |^ g & h in A
  proof
    assume x in A |^ g;
    then consider a,b such that
A1: x = a |^ b & a in A and
A2: b in {g};
    b = g by A2,TARSKI:def 1;
    hence thesis by A1;
  end;
  given h such that
A3: x = h |^ g & h in A;
  g in {g} by TARSKI:def 1;
  hence thesis by A3;
end;
