
theorem Th24: :: Pminmin:
for R being transitive RelStr, x being Element of R, S being Subset of R
 st x is_minimal_in Lower S holds x is_minimal_in [#]R
proof
 let R be transitive RelStr, x be Element of R, S be Subset of R such that
A1: x is_minimal_in Lower S;
 set cR = the carrier of R;
A2: x in Lower S by A1,WAYBEL_4:56;
 assume not x is_minimal_in [#]R;
 then consider y being Element of R such that  y in cR and
A3: y < x by A2,WAYBEL_4:56;
 per cases by A2,XBOOLE_0:def 3;
 suppose A4: x in S;
    y <= x by A3;
   then y in downarrow S by A4,WAYBEL_0:def 15;
   then y in Lower S by XBOOLE_0:def 3;
  hence thesis by A1,A3,WAYBEL_4:56;
 end;
 suppose  x in downarrow S;
   then consider x99 being Element of R such that
A5: x <= x99 and
A6: x99 in S by WAYBEL_0:def 15;
   y <= x by A3; then
 y <= x99 by A5,YELLOW_0:def 2;
   then y in downarrow S by A6,WAYBEL_0:def 15;
   then y in Lower S by XBOOLE_0:def 3;
  hence contradiction by A1,A3,WAYBEL_4:56;
 end;
end;
