theorem Th50:
  o is associative implies o.:^2 is associative
proof
  assume
A1: o is associative;
  let x,y,z be Subset of D;
  thus (o.:^2).((o.:^2).(x,y),z) = (o.:^2).(o.:[:x,y:],z) by Th44
    .= o.:[:o.:[:x,y:],z:] by Th44
    .= o.:[:x,o.:[:y,z:]:] by A1,Th48
    .= (o.:^2).(x,o.:[:y,z:]) by Th44
    .= (o.:^2).(x,(o.:^2).(y,z)) by Th44;
end;
