
theorem Th8:
  for R being RelStr holds R is well_founded iff well_founded-Part
  R = the carrier of R
proof
  let R be RelStr;
  set r = the InternalRel of R, c = the carrier of R, wfp = well_founded-Part
  R;
  set IT = {S where S is Subset of R : S is well_founded lower};
  c c= c;
  then reconsider cs = c as Subset of R;
A1: wfp = union IT by Def4;
  hereby
A2: cs is lower;
    assume R is well_founded;
    then r is_well_founded_in cs;
    then cs is well_founded;
    then cs in IT by A2;
    then cs c= wfp by A1,ZFMISC_1:74;
    hence wfp = c;
  end;
  assume
A3: wfp = c;
  let Y be set;
  assume that
A4: Y c= c and
A5: Y <> {};
  consider y being object such that
A6: y in Y by A5,XBOOLE_0:def 1;
  consider YY being set such that
A7: y in YY and
A8: YY in IT by A1,A3,A4,A6,TARSKI:def 4;
  consider S being Subset of R such that
A9: YY = S and
A10: S is well_founded lower by A8;
  set YS = Y /\ S;
A11: r is_well_founded_in S by A10;
  YS c= S & YS <> {} by A6,A7,A9,XBOOLE_0:def 4;
  then consider a being object such that
A12: a in YS and
A13: r-Seg a misses YS by A11;
A14: a in Y by A12,XBOOLE_0:def 4;
  a in S by A12,XBOOLE_0:def 4;
  then
A15: r-Seg a /\ Y = r-Seg a /\ S /\ Y by A10,Th4,XBOOLE_1:28
    .= r-Seg a /\ YS by XBOOLE_1:16;
  r-Seg a /\ YS = {} by A13;
  then r-Seg a misses Y by A15;
  hence thesis by A14;
end;
