
theorem
  for L be non empty transitive RelStr for S be meet-closed non empty
Subset of L for x,y be Element of S st ex_inf_of {x,y},L holds ex_inf_of {x,y},
  subrelstr S & "/\"({x,y},subrelstr S) = "/\"({x,y},L)
proof
  let L be non empty transitive RelStr;
  let S be meet-closed non empty Subset of L;
  let x,y be Element of S;
A1: x is Element of subrelstr S by YELLOW_0:def 15;
A2: y is Element of subrelstr S by YELLOW_0:def 15;
  assume
A3: ex_inf_of {x,y},L;
  subrelstr S is meet-inheriting non empty full SubRelStr of L by Def1;
  then "/\"({x,y},L) in the carrier of subrelstr S by A1,A2,A3,YELLOW_0:def 16;
  hence thesis by A1,A2,A3,YELLOW_0:65;
end;
