reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;

theorem
  R well_orders field R iff R is well-ordering
proof
  thus R well_orders field R implies R is well-ordering
  proof
    assume R is_reflexive_in field R & R is_transitive_in field R & R
is_antisymmetric_in field R & R is_connected_in field R & R is_well_founded_in
    field R;
    hence R is reflexive & R is transitive & R is antisymmetric & R is
    connected & R is well_founded by RELAT_2:def 9,def 12,def 14,def 16;
  end;
  assume R is reflexive & R is transitive & R is antisymmetric & R is
  connected & R is well_founded;
  hence R is_reflexive_in field R & R is_transitive_in field R & R
is_antisymmetric_in field R & R is_connected_in field R & R is_well_founded_in
  field R by RELAT_2:def 9,def 12,def 14,def 16;
end;
