
theorem Th33:
  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_inf_of X, product J
  for i being Element of I holds (inf X).i = inf 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_inf_of X, product J;
  then for i being Element of I holds ex_inf_of pi(X,i), J.i by Th31;
  then consider f being Element of product J such that
A1: for i being Element of I holds f.i = inf 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 Lm2;
  inf X = f by A2,A3,YELLOW_0:31;
  hence thesis by A1;
end;
