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