reserve a,b for object, I,J for set;
reserve b for bag of I;
reserve R for asymmetric transitive non empty RelStr,
  a,b,c for bag of the carrier of R,
  x,y,z for Element of R;
reserve p for partition of b-'a, q for partition of b;

theorem Th30:
  q = <*a*>^p & q is ordered implies p is ordered
  proof
    assume Z1: q = <*a*>^p;
    assume Z2: q is ordered;
    hereby let m be bag of the carrier of R;
      assume
A0:   m in rng p;
      then
A1:   m in rng<*a*>\/rng p = rng q by Z1,XBOOLE_0:def 3,FINSEQ_1:31;
      let x be Element of R; assume
A2:   m.x > 0;
      m divides Sum p = b-'a divides b by A0,Th26,PART,Lem3;
      then b.x = m.x <= (b-'a).x <= b.x by A1,A2,Z2,PRE_POLY:def 11;
      hence m.x = (b-'a).x by XXREAL_0:1;
    end;
    hereby let m be bag of the carrier of R;
      assume m in rng p;
      then m in rng<*a*>\/rng p = rng q by Z1,FINSEQ_1:31,XBOOLE_0:def 3;
      hence for x,y being Element of R st m.x > 0 & m.y > 0 & x <> y
      holds x ## y by Z2;
    end;
    hereby let m be bag of the carrier of R;
      assume m in rng p;
      then m in rng<*a*>\/rng p = rng q by Z1,FINSEQ_1:31,XBOOLE_0:def 3;
      hence m <> EmptyBag the carrier of R by Z2;
    end;
    let i be Nat;
    assume
A1: i in dom p & i+1 in dom p;
A2: len <*a*> = 1 by FINSEQ_1:40;
    then
A3: q.(1+i) = p.i & q.(1+(i+1)) = p.(i+1) by Z1,A1,FINSEQ_1:def 7;
A4: 1+i in dom q & 1+i+1 = 1+(i+1) in dom q by Z1,A1,A2,FINSEQ_1:28;
    then
    q.(1+i) = q/.(1+i) & p/.i = p.i & q.(1+(i+1)) = q/.(1+i+1) &
    p.(i+1) = p/.(i+1) by A1,PARTFUN1:def 6;
    hence thesis by Z2,A3,A4;
  end;
