
theorem
  for L being with_infima Poset for S being meet-inheriting non empty
full SubRelStr of L for x,y being Element of S, a,b be Element of L st a = x &
  b = y holds x"/\"y = a"/\"b
proof
  let L be with_infima Poset;
  let S be meet-inheriting non empty full SubRelStr of L;
  let x,y be Element of S, a,b be Element of L such that
A1: a = x & b = y;
A2: ex_inf_of {a,b},L by Th21;
  then "/\"({x,y},L) in the carrier of S by A1,Def16;
  then
A3: "/\"({x,y},S) = "/\"({x,y},L) by A1,A2,Th63;
  a"/\"b = inf {a,b} by Th40;
  hence thesis by A1,A3,Th40;
end;
