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

theorem Th30:
  (Map2Rel L) * (Rel2Map L) = id dom (Rel2Map L)
proof
  set LS = (Map2Rel L) * (Rel2Map L);
  set R = id dom (Rel2Map L);
  dom LS = the carrier of InclPoset Aux L by FUNCT_2:def 1;
  then
A1: dom LS = Aux L by YELLOW_1:1;
  dom R = the carrier of InclPoset Aux L by FUNCT_2:def 1;
  then
A2: dom R = Aux L by YELLOW_1:1;
  for a being object st a in Aux L holds LS.a = R.a
  proof
    let a be object;
    assume
A3: a in Aux L;
    then reconsider AR = a as auxiliary Relation of L by Def8;
A4: a in the carrier of InclPoset Aux L by A3,YELLOW_1:1;
    then
A5: a in dom (Rel2Map L) by FUNCT_2:def 1;
    then LS.a = (Map2Rel L).((Rel2Map L).AR) by FUNCT_1:13;
    then
A6: LS.a = (Map2Rel L).(AR-below) by Def14;
    LS.a in the carrier of InclPoset Aux L by A4,FUNCT_2:5;
    then LS.a in Aux L by YELLOW_1:1;
    then reconsider La = LS.a as auxiliary Relation of L by Def8;
A7: AR-below in the carrier of MonSet L by Th18;
    for c,b be object holds [c,b] in La iff [c,b] in AR
    proof
      let c,b be object;
      hereby
        assume
A8:     [c,b] in La;
        ex AR9 be auxiliary Relation of L st ( AR9 = (Map2Rel L).(
AR-below))&
(for c,b be object holds [c,b] in AR9 iff ex c9,b9 be Element of L, s9
be Function of L, InclPoset Ids L st c9 = c & b9 = b & s9 = AR-below & c9 in s9
        .b9) by A7,Def15;
        then consider c9,b9 be Element of L,
        s9 be Function of L, InclPoset Ids L such that
A9:     c9 = c and
A10:    b9 = b and
A11:    s9 = AR-below and
A12:    c9 in s9.b9 by A6,A8;
        c9 in AR-below b9 by A11,A12,Def12;
        hence [c,b] in AR by A9,A10,Th13;
      end;
      assume
A13:  [c,b] in AR;
      then reconsider c9 = c, b9 = b as Element of L by ZFMISC_1:87;
      c9 in AR-below b9 by A13;
      then
A14:  c9 in (AR-below).b9 by Def12;
      ex AR9 be auxiliary Relation of L st ( AR9 = (Map2Rel L).(
AR-below))&
(for c,b be object holds [c,b] in AR9 iff ex c9,b9 be Element of L, s9
be Function of L, InclPoset Ids L st c9 = c & b9 = b & s9 = AR-below & c9 in s9
      .b9) by A7,Def15;
      hence thesis by A6,A14;
    end;
    then La = AR;
    hence thesis by A5,FUNCT_1:18;
  end;
  hence thesis by A1,A2,FUNCT_1:2;
end;
