theorem Th4:
  for i st i in I holds (X (\+\) Y).i = X.i \+\ Y.i
proof
  let i such that
A1: i in I;
  thus (X (\+\) Y).i = (X (\) Y).i \/ (Y (\) X).i by A1,Def4
    .= X.i \ Y.i \/ (Y (\) X).i by A1,Def6
    .= X.i \+\ Y.i by A1,Def6;
end;
