
theorem Th51:
  for L be non empty reflexive transitive RelStr for S be non
  empty full SubRelStr of L holds dom supMap S = Ids S & rng supMap S is Subset
  of 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(supMap 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;
end;
