
theorem Th16: :: Def 4.30 (see WAYBEL_4:def 26)
  for L being RelStr, N being set, x being Element of L\~
  holds x is_minimal_wrt N, the InternalRel of (L\~) iff (x in N &
  for y being Element of L st y in N & [y,x] in the InternalRel of L
  holds [x,y] in the InternalRel of L)
proof
  let L be RelStr, N be set, x be Element of L\~;
  set IR = the InternalRel of L;
  set IR9 = the InternalRel of L\~;
  hereby
    assume
A1: x is_minimal_wrt N, the InternalRel of (L\~);
    hence x in N by WAYBEL_4:def 25;
    let y be Element of L such that
A2: y in N;
    assume
A3: [y,x] in IR;
    now per cases;
      suppose x = y;
        hence [x,y] in IR by A3;
      end;
      suppose x <> y;
        then not [y,x] in IR9 by A1,A2,WAYBEL_4:def 25;
        then [y,x] in IR~ by A3,XBOOLE_0:def 5;
        hence [x,y] in IR by RELAT_1:def 7;
      end;
    end;
    hence [x,y] in the InternalRel of L;
  end;
  assume that
A4: x in N and
A5: for y being Element of L st y in N holds [y,x] in (the InternalRel
  of L) implies [x,y] in (the InternalRel of L);
  now
    assume ex y being set st y in N & y <> x & [y,x] in IR9;
    then consider y being set such that
A6: y in N and y <> x and
A7: [y,x] in IR9;
    reconsider y9=y as Element of L by A7,ZFMISC_1:87;
A8: not [y,x] in IR~ by A7,XBOOLE_0:def 5;
    [y9,x] in IR implies [x,y9] in IR by A5,A6;
    hence contradiction by A7,A8,RELAT_1:def 7;
  end;
  hence thesis by A4,WAYBEL_4:def 25;
end;
