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;

theorem Th28:
  for p being Bags I-valued FinSequence
  for a,b being bag of I holds
  <*a*>^p is partition of b iff a divides b & p is partition of b-'a
  proof
    let p be Bags I-valued FinSequence;
    let a,b be bag of I;
    thus <*a*>^p is partition of b implies a divides b & p is partition of b-'a
    proof
      assume b = Sum(<*a*>^p);
      then A1: b = a+Sum p by Th25;
      hence a divides b by PRE_POLY:50;
      thus b-'a = Sum p by A1,PRE_POLY:48;
    end;
    assume Z1: a divides b;
    assume b-'a = Sum p;
    then Sum p+a = b-'a+a = b by Z1,PRE_POLY:47;
    hence b = Sum(<*a*>^p) by Th25;
  end;
