
theorem Th30:
  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 antisymmetric
  holds product J is antisymmetric
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 antisymmetric;
  let x,y be Element of product J such that
A2: x <= y and
A3: y <= x;
A4: dom x = I by Th27;
A5: dom y = I by Th27;
  now
    let j be object;
    assume j in I;
    then reconsider i = j as Element of I;
A6: x.i <= y.i by A2,Th28;
A7: y.i <= x.i by A3,Th28;
    J.i is antisymmetric by A1;
    hence x.j = y.j by A6,A7;
  end;
  hence thesis by A4,A5,FUNCT_1:2;
end;
