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 Th42:
  x in g |^ A iff ex h st x = g |^ h & h in A
proof
  thus x in g |^ A implies ex h st x = g |^ h & h in A
  proof
    assume x in g |^ A;
    then consider a,b such that
A1: x = a |^ b and
A2: a in {g} and
A3: b in A;
    a = g by A2,TARSKI:def 1;
    hence thesis by A1,A3;
  end;
  given h such that
A4: x = g |^ h & h in A;
  g in {g} by TARSKI:def 1;
  hence thesis by A4;
end;
