
theorem Th53:
  for L be non empty reflexive transitive RelStr for S be non
  empty full SubRelStr of L holds dom idsMap S = Ids S & rng idsMap S is Subset
  of Ids L
proof
  let L be non empty reflexive transitive RelStr;
  let S be non empty full SubRelStr of L;
  set P = InclPoset Ids S;
  thus dom(idsMap S) = the carrier of P by FUNCT_2:def 1
    .= the carrier of RelStr(#Ids S, RelIncl Ids S#) by YELLOW_1:def 1
    .= Ids S;
  thus thesis by YELLOW_1:1;
end;
