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;
