reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;
reserve f for Choice_Function of BOOL(the carrier of A);
reserve fC,fC1,fC2 for Chain of f;
reserve R for Relation,
  A for non empty Poset,
  C for Chain of A,
  S for Subset of A,
  a,a1,a2,b,c1,c2 for Element of A;

theorem
  (the InternalRel of A) |_2 C is being_linear-order
proof
  set P = (the InternalRel of A) |_2 C;
A1: P is_connected_in C
  proof
    let x,y be object;
    assume
A2: x in C & y in C;
    then
A3: [x,y] in [:C,C:] & [y,x] in [:C,C:] by ZFMISC_1:87;
    the InternalRel of A is_strongly_connected_in C by Def7;
    then
    [x,y] in the InternalRel of A or [y,x] in the InternalRel of A by A2;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  the InternalRel of A is being_partial-order;
  then P is being_partial-order by ORDERS_1:26;
  hence P is reflexive & P is transitive & P is antisymmetric;
  C = field P by Th46;
  hence thesis by A1;
end;
