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

theorem Sat7Serial: :: Unexpected as seriality can be then proven
  for R being finite non empty RelStr st
    R is satisfying(7H') holds R is serial
  proof
    let R be finite non empty RelStr;
    set U = UAp R;
    assume
tr: R is satisfying(7H');
    (UAp {}R)` = [#]R; then
Y2: [#]R c= UAp ([#]R) by tr;
FF: U.[#]R = UAp [#]R by ROUGHS_2:def 11
       .= the carrier of R by Y2,XBOOLE_0:def 10;
FO: U.{} = UAp {}R by ROUGHS_2:def 11 .= {};
    for X, Y being Subset of R holds U.(X \/ Y) = U.X \/ U.Y
    proof
      let X,Y be Subset of R;
      U.(X \/ Y) = UAp (X \/ Y) by ROUGHS_2:def 11
         .= (UAp X) \/ UAp Y by ROUGHS_2:13
         .= U.X \/ UAp Y by ROUGHS_2:def 11
         .= U.X \/ U.Y by ROUGHS_2:def 11;
      hence thesis;
    end; then
    consider S being non empty finite serial RelStr such that
H1: the carrier of S = the carrier of R & U = UAp S by ROUGHS_2:32,FF,FO;
    the RelStr of S = the RelStr of R by Corr3A,H1;
    hence thesis by NatDay;
  end;
