
theorem Th29:
  for I being non empty set
  for J being RelStr-yielding non-Empty ManySortedSet of I
  st for i being Element of I holds J.i is transitive
  holds product J is transitive
proof
  let I be non empty set;
  let J be RelStr-yielding non-Empty ManySortedSet of I such that
A1: for i being Element of I holds J.i is transitive;
  let x,y,z be Element of product J such that
A2: x <= y and
A3: y <= z;
  now
    let i be Element of I;
A4: x.i <= y.i by A2,Th28;
A5: y.i <= z.i by A3,Th28;
    J.i is transitive by A1;
    hence x.i <= z.i by A4,A5;
  end;
  hence thesis by Th28;
end;
