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 linearly_orders C
proof
A1: the InternalRel of A is_antisymmetric_in the carrier of A by Def4;
  the InternalRel of A is_reflexive_in the carrier of A & the InternalRel
  of A is_transitive_in the carrier of A by Def2,Def3;
  hence the InternalRel of A is_reflexive_in C & the InternalRel of A
  is_transitive_in C & the InternalRel of A is_antisymmetric_in C by A1;
  the InternalRel of A is_strongly_connected_in C by Def7;
  hence thesis;
end;
