reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th32:
  (Rel2Map L)" = Map2Rel L
proof
  (Rel2Map L)*(Map2Rel L) = id (the carrier of MonSet L) implies
  rng (Rel2Map L) = the carrier of MonSet L by FUNCT_2:18;
  then
A1: rng (Rel2Map L) = dom (Map2Rel L) by Th31,FUNCT_2:def 1;
  (Map2Rel L)*(Rel2Map L) = id dom (Rel2Map L) by Th30;
  hence thesis by A1,FUNCT_1:41;
end;
