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,b} |^ {c,d} = {a |^ c,a |^ d,b |^ c,b |^ d}
proof
  thus {a,b} |^ {c,d} c= {a |^ c,a |^ d,b |^ c,b |^ d}
  proof
    let x be object;
    assume x in {a,b} |^ {c,d};
    then consider g,h such that
A1: x = g |^ h and
A2: g in {a,b} and
A3: h in {c,d};
A4: h = c or h = d by A3,TARSKI:def 2;
    g = a or g = b by A2,TARSKI:def 2;
    hence thesis by A1,A4,ENUMSET1:def 2;
  end;
  let x be object;
A5: c in {c,d} & d in {c,d} by TARSKI:def 2;
  assume x in {a |^ c,a |^ d,b |^ c,b |^ d};
  then
A6: x = a |^ c or x = a |^ d or x = b |^ c or x = b |^ d by ENUMSET1:def 2;
  a in {a,b} & b in {a,b} by TARSKI:def 2;
  hence thesis by A6,A5;
end;
