reserve R for Ring,
  r for Element of R,
  a, b, d1, d2 for Data-Location of R,
  il, i1, i2 for Nat,
  I for Instruction of SCM R,
  s,s1, s2 for State of SCM R,
  T for InsType of the InstructionsF of SCM R,
  k for Nat;

theorem
  for R being non trivial Ring, a being Data-Location of R, il, i1
being Element of NAT holds NIC(a=0_goto i1, il) = {i1, il + 1}
proof
  let R be non trivial Ring, a be Data-Location of R, il, i1 be
  Element of NAT;
  set t = the State of SCM R,
      Q = the Instruction-Sequence of SCM R;
  set I = a=0_goto i1;
  reconsider a9 = a as Element of Data-Locations SCM by SCMRING2:1;
A1: Values a = ((SCM-VAL R)*SCM-OK).a9 by SCMRING2:24
    .= the carrier of R by AMI_3:27,SCMRING1:4;
  reconsider il1 = il as Element of Values IC SCM R by MEMSTR_0:def 6;
  thus NIC(a=0_goto i1, il) c= {i1, il + 1} by Th31;
  reconsider u = t+*(IC SCM R,il1)
   as Element of product the_Values_of SCM R by CARD_3:107;
  reconsider P = Q +* (il,I) as Instruction-Sequence of SCM R;
  let x be object;
A2: IC SCM R <> a by Th2;
A3: IC SCM R in dom t by MEMSTR_0:2;
  assume
A4: x in {i1, il + 1};
  per cases by A4,TARSKI:def 2;
  suppose
A5: x = i1;
    reconsider 0R = 0.R as Element of Values a by A1;
    reconsider v = u+*(a .--> 0R)
     as Element of product the_Values_of SCM R by CARD_3:107;
    not IC SCM R in dom (a .--> 0R) by A2,TARSKI:def 1;
    then
A7: IC v = IC u by FUNCT_4:11
      .= il by A3,FUNCT_7:31;
A8:   P/.il = P.il by PBOOLE:143;
    il in NAT;
    then il in dom Q by PARTFUN1:def 2;
    then
A9: P.il = I by FUNCT_7:31;
    a in dom (a .--> 0R) by TARSKI:def 1;
    then v.a = (a .--> 0R).a by FUNCT_4:13
      .= 0.R by FUNCOP_1:72;
    then IC Following(P,v) = i1 by A7,A8,A9,SCMRING2:16;
    hence thesis by A5,A7,A8,A9;
  end;
  suppose
A10: x = il + 1;
    consider e being Element of R such that
A11: e <> 0.R by STRUCT_0:def 18;
    reconsider E = e as Element of Values a by A1;
    reconsider v = u+*(a .--> E)
     as Element of product the_Values_of SCM R by CARD_3:107;
    not IC SCM R in dom (a .--> E) by A2,TARSKI:def 1;
    then
A13: IC v = IC u by FUNCT_4:11
      .= il by A3,FUNCT_7:31;
A14:   P/.il = P.il by PBOOLE:143;
    il in NAT;
    then il in dom Q by PARTFUN1:def 2;
    then
A15: P.il = I by FUNCT_7:31;
    a in dom (a .--> E) by TARSKI:def 1;
    then v.a = (a .--> E).a by FUNCT_4:13
      .= E by FUNCOP_1:72;
    then IC Following(P,v) = il + 1 by A11,A13,A14,A15,SCMRING2:16;
    hence thesis by A10,A13,A14,A15;
  end;
end;
