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

theorem Th10:
   (BagN1)*(NBag1) = id NAT
   proof
     for o st o in dom ((BagN1)*NBag1) holds ((BagN1)*(NBag1)).o = (id NAT).o
     proof
       let o;
       assume
A1:    o in dom ((BagN1)*NBag1); then
       reconsider m = o as Element of NAT;
A2:    NBag1.o = 1 --> m by Def1;
       reconsider b = (NBag1).o as Element of Bags 1 by A2,PRE_POLY:def 12;
A3:    1 --> m = b by Def1;
       0 in 1 by CARD_1:49,TARSKI:def 1; then
A4:    m = b.0 by A3,FUNCOP_1:7;
       ((BagN1)*NBag1).o = (BagN1).b by A1,FUNCT_2:15
       .= o by A4,Def2
       .= (id NAT).o by A1,FUNCT_1:18;
       hence thesis;
     end;
     hence thesis by FUNCT_2:def 1;
   end;
