reserve A, B, X, Y, Z, x, y for set;
reserve f for Function;
reserve O for Ordinal;

theorem
  for F being set st F is finite & F <> {} & F is c=-linear
  ex m being set st m in F & for C being set st C in F holds m c= C
proof
  defpred P[set] means $1 <> {} implies
  ex m being set st m in $1 & for C being set st C in $1 holds m c= C;
  let F be set such that
A1: F is finite and
A2: F <> {} and
A3: F is c=-linear;
A4: P[{}];
A5: now
    let x,B be set such that
A6: x in F and
A7: B c= F and
A8: P[B];
    now per cases;
      suppose
A9:     not ex y being set st y in B & y c=x;
        assume B \/ {x} <> {};
        take m = x;
        x in {x} by TARSKI:def 1;
        hence m in B \/ {x} by XBOOLE_0:def 3;
        let C be set;
        assume C in B \/ {x};
        then
A10:    C in B or C in {x} by XBOOLE_0:def 3;
        then C,x are_c=-comparable by A3,A6,A7,TARSKI:def 1;
        then C c= x or x c= C;
        hence m c= C by A9,A10,TARSKI:def 1;
      end;
      suppose ex y being set st y in B & y c=x;
        then consider y being set such that
A11:    y in B and
A12:    y c=x;
        assume B \/ {x} <> {};
        consider m being set such that
A13:    m in B and
A14:    for C being set st C in B holds m c= C by A8,A11;
        m c= y by A11,A14;
        then
A15:    m c= x by A12;
        take m;
        thus m in B \/ {x} by A13,XBOOLE_0:def 3;
        let C be set;
        assume C in B \/ {x};
        then C in B or C in {x} by XBOOLE_0:def 3;
        hence m c= C by A14,A15,TARSKI:def 1;
      end;
    end;
    hence P[B \/ {x}];
  end;
  P[F] from Finite(A1,A4,A5);
  hence thesis by A2;
end;
