
theorem Th21:
  for L be non empty transitive RelStr for S be infs-closed non
  empty Subset of L for X be Subset of S st ex_inf_of X,L holds ex_inf_of X,
  subrelstr S & "/\"(X,subrelstr S) = "/\"(X,L)
proof
  let L be non empty transitive RelStr;
  let S be infs-closed non empty Subset of L;
  let X be Subset of S;
A1: X is Subset of subrelstr S by YELLOW_0:def 15;
  assume
A2: ex_inf_of X,L;
  subrelstr S is infs-inheriting non empty full SubRelStr of L by Def3;
  then "/\"(X,L) in the carrier of subrelstr S by A1,A2,YELLOW_0:def 18;
  hence thesis by A1,A2,YELLOW_0:63;
end;
