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} = {a |^ b,a |^ c}
proof
  thus {a} |^ {b,c} c= {a |^ b,a |^ c}
  proof
    let x be object;
    assume x in {a} |^ {b,c};
    then consider g,h such that
A1: x = g |^ h and
A2: g in {a} and
A3: h in{b,c};
A4: h = b or h = c by A3,TARSKI:def 2;
    g = a by A2,TARSKI:def 1;
    hence thesis by A1,A4,TARSKI:def 2;
  end;
  let x be object;
A5: c in {b,c} by TARSKI:def 2;
  assume x in {a |^ b,a |^ c};
  then
A6: x = a |^ b or x = a |^ c by TARSKI:def 2;
  a in {a} & b in {b,c} by TARSKI:def 1,def 2;
  hence thesis by A6,A5;
end;
