
theorem :: Lemma 4.38
  for R being non empty Poset
  holds the InternalRel of R well_orders the carrier of R iff
  for N being non empty Subset of R holds card min-classes N = 1
proof
  let R be non empty Poset;
  set IR = the InternalRel of R, CR = the carrier of R;
A1: R is quasi_ordered;
  hereby
    assume
A2: IR well_orders CR;
    then
A3: IR is_reflexive_in CR by WELLORD1:def 5;
A4: IR is_connected_in CR by A2,WELLORD1:def 5;
A5: IR is_well_founded_in CR by A2,WELLORD1:def 5;
    IR is_strongly_connected_in CR by A3,A4,ORDERS_1:7;
    then
A6: R is connected by Th4;
    R is well_founded by A5,WELLFND1:def 2;
    then R\~ is well_founded by Th14;
    hence for N being non empty Subset of R holds
    card min-classes N = 1 by A1,A6,Th22;
  end;
  assume
A7: for N being non empty Subset of R holds card min-classes N = 1;
  then
A8: R is connected by A1,Th22;
A9: R\~ is well_founded by A1,A7,Th22;
A10: IR is_strongly_connected_in CR by A8,Th4;
A11: R is well_founded by A9,Th15;
A12: IR is_reflexive_in CR by ORDERS_2:def 2;
A13: IR is_transitive_in CR by ORDERS_2:def 3;
A14: IR is_antisymmetric_in CR by ORDERS_2:def 4;
A15: IR is_connected_in CR by A10;
  IR is_well_founded_in CR by A11,WELLFND1:def 2;
  hence thesis by A12,A13,A14,A15,WELLORD1:def 5;
end;
