
theorem Th37:  :: Proposition 5 1H 4H 3H
  for A being non empty finite set,
      U being Function of bool A, bool A st
    U.{} = {} &
    (for X being Subset of A holds X c= U.X) &
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y) holds
  ex R being non empty finite reflexive RelStr st
    the carrier of R = A & U = UAp R
  proof
    let A be non empty finite set;
    let U be Function of bool A, bool A;
    assume that
A1: U.{} = {} and
A2: for X being Subset of A holds X c= U.X and
A3: for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y;
    U.([#]A) c= [#]A & [#]A c= U.([#]A) by A2; then
    U.A = A; then
    consider R being non empty finite serial RelStr such that
A4: the carrier of R = A & U = UAp R by Th32,A1,A3;
    for x being object st x in the carrier of R holds
      [x,x] in the InternalRel of R
    proof
      let x be object;
      assume
A5:    x in the carrier of R; then
A6:   {x} is Subset of R by ZFMISC_1:31;
      reconsider Z = {x} as Subset of R by A5,ZFMISC_1:31;
A7:   {x} c= U.({x}) by A4,A6,A2;
      x in {x} by TARSKI:def 1; then
A8:   x in U.({x}) by A7;
      U.({x}) = UAp Z by Def11,A4
             .= {y where y is Element of R :
             Class (the InternalRel of R,y) meets Z}; then
      consider t being Element of R such that
A9:   t = x & Class (the InternalRel of R,t) meets Z by A8;
      x in Class (the InternalRel of R,t)
      proof
        assume
A10:     not x in Class (the InternalRel of R,t);
        consider y being object such that
A11:     y in Class (the InternalRel of R,t) /\ {x} by A9,XBOOLE_0:def 1;
        y in Class (the InternalRel of R,t) & y in {x} by XBOOLE_0:def 4,A11;
        hence contradiction by A10,TARSKI:def 1;
      end;
      hence thesis by A9,RELAT_1:169;
    end;
    then R is reflexive by ORDERS_2:def 2,RELAT_2:def 1;
    hence thesis by A4;
  end;
