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 S is being_linear-order implies S is Chain of A
proof
  assume (the InternalRel of A) |_2 S is being_linear-order;
  then
A1: (the InternalRel of A) |_2 S is connected;
  field((the InternalRel of A) |_2 S) = S by Th46;
  then
A2: (the InternalRel of A) |_2 S is_connected_in S by A1;
  S is strongly_connected
  proof
    let x,y be object;
    assume
A3: x in S & y in S;
    then reconsider a = x, b = y as Element of A;
    now
      per cases;
      suppose
        x = y;
        then a <= b;
        hence thesis;
      end;
      suppose
        x <> y;
        then [x,y] in (the InternalRel of A) |_2 S or [y,x] in (the
        InternalRel of A) |_2 S by A2,A3;
        hence thesis by XBOOLE_0:def 4;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
