
theorem Th32:
  for I being non empty set for J being Poset-yielding non-Empty
  ManySortedSet of I for X being Subset of product J st ex_sup_of X, product J
  for i being Element of I holds (sup X).i = sup pi(X,i)
proof
  let I be non empty set;
  let J be Poset-yielding non-Empty ManySortedSet of I;
  let X be Subset of product J;
  assume ex_sup_of X, product J;
  then for i being Element of I holds ex_sup_of pi(X,i), J.i by Th30;
  then consider f being Element of product J such that
A1: for i being Element of I holds f.i = sup pi(X,i) and
A2: f is_>=_than X and
A3: for g being Element of product J st X is_<=_than g holds f <= g by Lm1;
  sup X = f by A2,A3,YELLOW_0:30;
  hence thesis by A1;
end;
