 reserve o,o1,o2 for object;
 reserve n for Ordinal;
 reserve R,L for non degenerated comRing;
 reserve b for bag of 1;

theorem Th14:
   for b be bag of 1 holds
   rng(NBag1|(Segm ((b.0)+1))) = {x where x is bag of 1 : x.0 <= b.0}
   proof
     let b be bag of 1;
A1:  for o holds o in rng(NBag1|(Segm ((b.0)+1))) implies
     o in {x where x is bag of 1 : x.0 <= b.0}
     proof
       let o;
       assume o in rng(NBag1|(Segm ((b.0)+1))); then
       consider x being object such that
A2:    x in dom(NBag1|(Segm ((b.0)+1))) & o = (NBag1|(Segm ((b.0)+1))).x
         by FUNCT_1:def 3;
       reconsider m1 = x as Element of NAT by A2;
       m1 < (b.0)+1 by A2, NAT_1:44; then
A3:    m1 <= b.0 by NAT_1:13;
A4:    (NBag1|(Segm ((b.0)+1))).x = (NBag1).x by A2,FUNCT_1:47
       .= 1--> m1 by Def1;
       0 in 1 by CARD_1:49,TARSKI:def 1; then
       (1--> m1).0 = m1 by FUNCOP_1:7;
       hence thesis by A3, A4,A2;
     end;
     for o holds o in {x where x is bag of 1 : x.0 <= b.0}
     implies o in rng(NBag1|(Segm ((b.0)+1)))
     proof
       let o;
       assume o in {x where x is bag of 1 : x.0 <= b.0}; then
       consider x1 be bag of 1 such that
A5:    o = x1 & x1.0 <= b.0;
       dom x1 = {0} by CARD_1:49, PARTFUN1:def 2; then
       rng x1 = {x1.0} by FUNCT_1:4; then
A6:   x1 = (dom x1) --> x1.0 by FUNCOP_1:9
       .= 1 --> x1.0 by PARTFUN1:def 2;
       reconsider m1 = x1.0 as Element of NAT;
       m1 < b.0 + 1 by A5,NAT_1:13; then
A7:   m1 in dom(NBag1|(Segm ((b.0)+1))) by NAT_1:44; then
       (NBag1|(Segm ((b.0)+1))).m1 = (NBag1).m1 by FUNCT_1:47
       .= 1--> m1 by Def1;
       hence thesis by A5,A6,A7,FUNCT_1:def 3;
     end;
     hence thesis by A1,TARSKI:2;
   end;
