 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem :: Theorem 1 L
  for A being non empty finite set,
      L being Function of bool A, bool A st
    L.A = A &
    (for X being Subset of A holds L.X = L.(L.X)) &
    (for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y) holds
  ex R being non empty mediate finite transitive RelStr st
    the carrier of R = A & L = LAp R
  proof
    let A be non empty finite set;
    let L be Function of bool A, bool A;
    assume
A0: L.A = A &
    (for X being Subset of A holds L.X = L.(L.X)) &
    (for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y); then
    for X being Subset of A holds L.X c= L.(L.X); then
    consider R being non empty finite transitive RelStr such that
A1: the carrier of R = A & L = LAp R by ThProposition9,A0;
    for X being Subset of A holds L.(L.X) c= L.X by A0; then
    consider R2 being non empty mediate finite RelStr such that
A2: the carrier of R2 = A & L = LAp R2 by ROUGHS_2:42,A0;
    reconsider LL = L as Function of bool the carrier of R2,
      bool the carrier of R2 by A2;
    set H = Flip LL;
F1: H.{} = {} by ROUGHS_2:19,A0,A2;
f2: for S, T being Subset of the carrier of R2
      holds H.(S \/ T) = H.S \/ H.T by ROUGHS_2:22,A2,A0;
    set Rx = GeneratedRelStr H;
S2: UAp R2 = H by ROUGHS_2:28,A2;
S3: UAp Rx = H by KeyTheorem,F1,f2,ROUGHS_4:def 9; then
S5: the InternalRel of Rx = the InternalRel of R2 by S2,Corr3A;
    H = UAp R by A1,ROUGHS_2:28,A2; then
    the InternalRel of Rx = the InternalRel of R by Corr3A,S3,A1,A2; then
    reconsider RRR = the RelStr of Rx as
      non empty finite mediate transitive RelStr by S5,Th13,A2,A1;
    take RRR;
    Flip UAp RRR = LAp RRR by ROUGHS_2:27;
    hence thesis by ROUGHS_2:27,A2,S3,S2;
  end;
