reserve U for Universe;

theorem Th4:
  for X being set st X in fin_RelStr_sp holds X is finite strict
  non empty RelStr
proof
  let X be set;
  assume
A1: X in fin_RelStr_sp;
  then
A2: ex R being strict RelStr st X = R & the carrier of R in FinSETS by Def4;
  then reconsider R=X as strict RelStr;
  now
    set M = fin_RelStr_sp \ {R}, F = fin_RelStr_sp;
    reconsider M as Subset of fin_RelStr;
A3: R in {R} by TARSKI:def 1;
    assume
A4: R is empty;
A5: now
      let H1,H2 be strict RelStr;
      assume that
A6:   (the carrier of H1) misses (the carrier of H2) and
A7:   H1 in M and
A8:   H2 in M;
A9:   H2 in F by A8,XBOOLE_0:def 5;
A10:  not H1 in {R} by A7,XBOOLE_0:def 5;
      the carrier of H1 <> {}
      proof
        per cases by A10,TARSKI:def 1;
        suppose
          (the carrier of H1) <> (the carrier of R);
          hence thesis by A4;
        end;
        suppose
A11:      (the InternalRel of H1) <> (the InternalRel of R);
          set InterH1 = the InternalRel of H1;
          InterH1 <> {} by A4,A11;
          hence thesis;
        end;
      end;
      then reconsider A = the carrier of H1 as non empty set;
      A \/ (the carrier of H2) <> {};
      then union_of(H1,H2) <> R by A4,Def2;
      then
A12:  not union_of(H1,H2) in {R} by TARSKI:def 1;
      the carrier of sum_of(H1,H2) = A \/ (the carrier of H2) by Def3;
      then
A13:  not sum_of(H1,H2) in {R} by A4,TARSKI:def 1;
A14:  H1 in F by A7,XBOOLE_0:def 5;
      then union_of(H1,H2) in F by A6,A9,Def5;
      hence union_of(H1,H2) in M by A12,XBOOLE_0:def 5;
      sum_of(H1,H2) in F by A6,A14,A9,Def5;
      hence sum_of(H1,H2) in M by A13,XBOOLE_0:def 5;
    end;
    now
      let S be strict RelStr;
      assume that
A15:  the carrier of S is 1-element and
A16:  the carrier of S in FinSETS;
A17:  not S in {R} by A4,A15,TARSKI:def 1;
      S in F by A15,A16,Def5;
      hence S in M by A17,XBOOLE_0:def 5;
    end;
    then F c= M by A5,Def5;
    hence contradiction by A1,A3,XBOOLE_0:def 5;
  end;
  hence thesis by A2;
end;
