
theorem Th28:
  for I being non empty set for J being Poset-yielding non-Empty
ManySortedSet of I for f being Element of product J, X being Subset of product
J holds f is_>=_than X iff for i being Element of I holds f.i is_>=_than pi(X,i
  )
proof
  let I be non empty set;
  let J be Poset-yielding non-Empty ManySortedSet of I;
  let f be Element of product J, X be Subset of product J;
  hereby
    assume
A1: f is_>=_than X;
    let i be Element of I;
    thus f.i is_>=_than pi(X, i)
    proof
      let x be Element of J.i;
      assume x in pi(X, i);
      then consider g being Function such that
A2:   g in X and
A3:   x = g.i by CARD_3:def 6;
      reconsider g as Element of product J by A2;
      g <= f by A1,A2;
      hence thesis by A3,WAYBEL_3:28;
    end;
  end;
  assume
A4: for i being Element of I holds f.i is_>=_than pi(X,i);
  let g be Element of product J;
  assume
A5: g in X;
  now
    let i be Element of I;
A6: f.i is_>=_than pi(X,i) by A4;
    g.i in pi(X,i) by A5,CARD_3:def 6;
    hence g.i <= f.i by A6;
  end;
  hence thesis by WAYBEL_3:28;
end;
