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;
