reserve i,j for Nat;
reserve A,B for Ring;

theorem Th1:
  for L1,L2,L3 be Ring st L1 is Subring of L2 & L2 is Subring of L3 holds
  L1 is Subring of L3
  proof
   let L1,L2,L3 be Ring;
   assume that
A1:  L1 is Subring of L2 and
A2:  L2 is Subring of L3;
A3:  the carrier of L1 c= the carrier of L2
  & the addF of L1 = (the addF of L2) || the carrier of L1
  & the multF of L1 = (the multF of L2) || the carrier of L1
  & 1.L1 = 1.L2& 0.L1 = 0.L2 by A1,C0SP1:def 3;
A4: the carrier of L2 c= the carrier of L3
  & the addF of L2= (the addF of L3) || the carrier of L2
  & the multF of L2= (the multF of L3) || the carrier of L2
  & 1.L2= 1.L3 & 0.L2= 0.L3 by A2,C0SP1:def 3;
  set A1 = [:the carrier of L1, the carrier of L1:];
  set A2 = [:the carrier of L2, the carrier of L2:];
  set A3 = [:the carrier of L3, the carrier of L3:];
A8: the carrier of L1 c= the carrier of L3 by A3,A4;
A9: the addF of L1 = (the addF of L2) || the carrier of L1
      by A1,C0SP1:def 3
    .= ((the addF of L3) || the carrier of L2) || the carrier of L1
      by A2,C0SP1:def 3
    .= (the addF of L3) || the carrier of L1 by A3,ZFMISC_1:96,RELAT_1:74;
A10:   the multF of L1 = (the multF of L2) || the carrier of L1
      by A1,C0SP1:def 3
      .= ((the multF of L3) || the carrier of L2) || the carrier of L1
      by A2,C0SP1:def 3
    .= (the multF of L3) || the carrier of L1 by A3,ZFMISC_1:96,RELAT_1:74;
A11: 1.L1 = 1.L2 by A1,C0SP1:def 3 .=1.L3 by A2,C0SP1:def 3;
     0.L1 = 0.L2 by A1,C0SP1:def 3 .=0.L3 by A2,C0SP1:def 3;
     hence thesis by A8,A9,A10,A11,C0SP1:def 3;
  end;
