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
  q is ordered & (q/.1).x = 0 & b.x > 0 implies
  ex y st (q/.1).y > 0 & x <= y
  proof
    assume Z0: q is ordered;
    assume Z2: (q/.1).x = 0;
    assume Z3: b.x > 0;
    defpred P[Nat] means $1 in dom q implies for x st (q/.$1).x > 0
    ex y st (q/.1).y > 0 & x <= y;
A1: P[2]
    proof
      assume
A2:   2 in dom q;
      then 2 <= len q by FINSEQ_3:25;
      then 1 <= len q by XXREAL_0:2;
      then 1 in dom q & 2 = 1+1 by FINSEQ_3:25;
      hence thesis by A2,Z0;
    end;
A3: for i being Nat st 2 <= i & P[i] holds P[i+1]
    proof let i be Nat;
      assume
A4:   2 <= i & P[i] & i+1 in dom q;
      then i <= i+1 <= len q by NAT_1:11,FINSEQ_3:25;
      then
A0:   1 <= i <= len q by A4,XXREAL_0:2;
      then
A5:   i in dom q by FINSEQ_3:25;
      let x; assume (q/.(i+1)).x > 0;
      then consider y such that
A6:   (q/.i).y > 0 & x <= y by Z0,A4,A5;
      consider z such that
A7:   (q/.1).z > 0 & y <= z by A4,A0,A6,FINSEQ_3:25;
      take z;
      thus (q/.1).z > 0 by A7;
      thus x <= z by A6,A7,ORDERS_2:3;
    end;
A8: for i being Nat st i >= 2 holds P[i] from NAT_1:sch 8(A1,A3);
    b = Sum q by PART;
    then consider i being Nat such that
A9: i in dom q & (q/.i).x > 0 by Z3,Lem14;
    1 <= i & i <> 1 by A9,Z2,FINSEQ_3:25;
    then 1 < i by XXREAL_0:1;
    then 1+1 <= i by NAT_1:13;
    hence thesis by A8,A9;
  end;
