
theorem Th34:
  for F being set st
    F is non empty with_non-empty_elements c=-linear holds F is centered
proof
  defpred P[set] means $1 <> {} implies ex x being set st x in $1 &
  for y being set st y in $1 holds x c= y;
  let F be set;
  assume that
A1: F is non empty and
A2: F is with_non-empty_elements and
A3: F is c=-linear;
  thus F <> {} by A1;
  let G be set;
  assume that
A4: G <> {} and
A5: G c= F and
A6: G is finite;
A7: now
    let x, B be set;
    assume that
A8: x in G and
A9: B c= G and
A10: P[B];
    thus P[B \/ {x}]
    proof
      assume B \/ {x} <> {};
      per cases;
      suppose
A11:    B is empty;
        take x9 = x;
        thus x9 in B \/ {x} by A11,TARSKI:def 1;
        let y be set;
        thus thesis by A11,TARSKI:def 1;
      end;
      suppose
    B is non empty;
        then consider z being set such that
A12:    z in B and
A13:    for y being set st y in B holds z c= y by A10;
        thus ex x9 being set st x9 in B \/ {x} & for y being set st y in B \/
        {x} holds x9 c= y
        proof
          z in G by A9,A12; then
A14:      x, z are_c=-comparable by A3,A5,A8,ORDINAL1:def 8;
          per cases by A14;
          suppose
A15:        x c= z;
            take x9 = x;
            x9 in {x} by TARSKI:def 1;
            hence x9 in B \/ {x} by XBOOLE_0:def 3;
            let y be set;
            assume
A16:        y in B \/ {x};
            per cases by A16,XBOOLE_0:def 3;
            suppose
              y in B;
              then z c= y by A13;
              hence thesis by A15;
            end;
            suppose
              y in {x};
              hence thesis by TARSKI:def 1;
            end;
          end;
          suppose
A17:        z c= x;
            take x9 = z;
            thus x9 in B \/ {x} by A12,XBOOLE_0:def 3;
            let y be set;
            assume
A18:        y in B \/ {x};
            per cases by A18,XBOOLE_0:def 3;
            suppose
              y in B;
              hence thesis by A13;
            end;
            suppose
              y in {x};
              hence thesis by A17,TARSKI:def 1;
            end;
          end;
        end;
      end;
    end;
  end;
A19: P[{}];
  P[G] from FINSET_1:sch 2(A6, A19, A7);
  then consider x being set such that
A20: x in G and
A21: for y being set st y in G holds x c= y by A4;
  consider xe being object such that
A22: xe in x by A2,A5,A20,XBOOLE_0:def 1;
  now
    let y be set;
    assume y in G;
    then x c= y by A21;
    hence xe in y by A22;
  end;
  hence thesis by A4,SETFAM_1:def 1;
end;
