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;
reserve J for set, m for bag of I;

theorem Lem14:
  for p being Bags I-valued FinSequence, x being object
  st x in I & (Sum p).x > 0
  ex i being Nat st i in dom p & (p/.i).x > 0
  proof
    let p be Bags I-valued FinSequence;
    let x be object;
    assume Z0: x in I;
    assume Z1: (Sum p).x > 0;
    defpred P[object] means for p being Bags I-valued FinSequence
    st p = $1 & (Sum p).x > 0 ex i being Nat st i in dom p & (p/.i).x > 0;
A1: P[ {} ]
    proof
      let p be Bags I-valued FinSequence;
      assume p = {};
      then p = <*>Bags I;
      then Sum p = EmptyBag I by Th21;
      hence thesis by Z0,FUNCOP_1:7;
    end;
A2: for p being FinSequence, a being object st P[p] holds P[p^<*a*>]
    proof
      let p be FinSequence;
      let a be object;
      assume Z0: P[p];
      let q be Bags I-valued FinSequence;
      assume
A3:   q = p^<*a*>;
      then reconsider p, aa = <*a*> as Bags I-valued FinSequence by Lem8;
      len aa = 1 by FINSEQ_1:40;
      then 1 in dom aa by FINSEQ_3:25;
      then aa.1 in Bags I by FUNCT_1:102;
      then reconsider a as Element of Bags I;
      assume (Sum q).x > 0;
      then (Sum p).x+a.x = ((Sum p)+a).x > 0 by Th22,A3,PRE_POLY:def 5;
      then per cases;
      suppose
A4:     a.x > 0;
        take i = len p+1;
        len q = len p+1 >= 1 by A3,NAT_1:11,FINSEQ_2:16;
        hence i in dom q by FINSEQ_3:25;
        then q/.i = q.i by PARTFUN1:def 6;
        hence thesis by A4,A3,FINSEQ_1:42;
      end;
      suppose
        (Sum p).x > 0;
        then consider i being Nat such that
A5:     i in dom p & (p/.i).x > 0 by Z0;
        take i;
A6:     dom p c= dom q by A3,FINSEQ_1:26;
        hence i in dom q by A5;
        q/.i = q.i = p.i by A3,A5,A6,FINSEQ_1:def 7,PARTFUN1:def 6;
        hence (q/.i).x > 0 by PARTFUN1:def 6,A5;
      end;
    end;
    for p being FinSequence holds P[p] from FINSEQ_1:sch 3(A1,A2);
    hence thesis by Z1;
  end;
