
theorem Th33:
  for A being transitive RelStr, B being finite Subset of A
    st B is non empty & the InternalRel of A is_connected_in B holds
      ex x being Element of A st x in B &
        for y being Element of A st y in B & x <> y holds
          y <= x
proof
  let A be transitive RelStr;
  let B be finite Subset of A;
  assume A1: B is non empty & the InternalRel of A is_connected_in B;
  defpred P[set] means
    $1 is non empty implies ex x being Element of A st x in $1 &
      for y being Element of A st y in $1 & x <> y holds
        y <= x;
  A2: B is finite;
  A3: P[{}];
  A4: for z, C being set st z in B & C c= B & P[C] holds P[C \/ {z}]
  proof
    let z, C be set;
    assume that
      A5: z in B and
      A6: C c= B and
      A7: P[C];
    set D = C \/ {z};
    per cases;
    suppose z in C;
      then D = C by ZFMISC_1:31, XBOOLE_1:12;
      hence thesis by A7;
    end;
    suppose A8: not z in C;
      ex x being Element of A st x in D &
        for y being Element of A st y in D & x <> y holds
          y <= x
      proof
        z in {z} by TARSKI:def 1;
        then A9: z in D by XBOOLE_0:def 3;
        consider x being Element of A such that
          A10: C is non empty implies x in C &
            for y being Element of A st y in C & x <> y holds
              y <= x by A7;
        per cases;
        suppose A11: C is empty;
          reconsider z as Element of A by A5;
          take z;
          thus z in D by A9;
          let y be Element of A;
          assume y in D & z <> y;
          hence thesis by A11, TARSKI:def 1;
        end;
        suppose A12: C is non empty;
          A13: A is non empty by A5;
          reconsider z as Element of A by A5;
          A14: x in B by A12, A10, A6;
          z <> x by A12, A8, A10;
          then per cases by A1, A13, A5, A14, Th26;
          suppose A15: x <= z;
            take z;
            thus z in D by A9;
            let y be Element of A;
            assume that
              A16: y in D and
              A17: z <> y;
            per cases;
            suppose y = x;
              hence y <= z by A15;
            end;
            suppose A18: y <> x;
              y in C by A16, A17, ZFMISC_1:136;
              then y <= x by A18, A10;
              hence y <= z by A15, ORDERS_2:3;
            end;
          end;
          suppose A19: z <= x;
            take x;
            thus x in D by A12, A10, XBOOLE_0:def 3;
            let y be Element of A;
            assume that
              A20: y in D and
              A21: x <> y;
            per cases;
            suppose y = z;
              hence y <= x by A19;
            end;
            suppose y <> z;
              then y in C by A20, ZFMISC_1:136;
              hence y <= x by A21, A10;
            end;
          end;
        end;
      end;
      hence P[D];
    end;
  end;
  P[B] from FINSET_1:sch 2(A2, A3, A4);
  hence thesis by A1;
end;
