reserve a, b, d1, d2, d3, d4 for Int-Location,
  A, B for Data-Location,
  f, f1, f2, f3 for FinSeq-Location,
  il, i1, i2 for Nat,
  L for Nat,
  I for Instruction of SCM+FSA,
  s,s1,s2 for State of SCM+FSA,
  T for InsType of the InstructionsF of SCM+FSA,
  k for Nat;
reserve J,K for Element of Segm 13,
  b,b1,c,c1 for Element of SCM-Data-Loc,
  f,f1 for Element of SCM+FSA-Data*-Loc;
reserve a, b, d1, d2, d3, d4 for Int-Location,
  A, B for Data-Location,
  f, f1,
  f2, f3 for FinSeq-Location;

theorem Th37:
  NIC(a>0_goto i1, il) = {i1, il + 1}
proof
  set t = the State of SCM+FSA,
  Q = the Instruction-Sequence of SCM+FSA;
  hereby
    let x be object;
    assume x in NIC(a>0_goto i1, il);
    then consider s being Element of product the_Values_of SCM+FSA
    such that
A1: x = IC Exec(a>0_goto i1,s) and
A2: IC s = il;
    per cases;
    suppose
      s.a > 0;
      then x = i1 by A1,SCMFSA_2:71;
      hence x in {i1, il + 1} by TARSKI:def 2;
    end;
    suppose
      s.a <= 0;
      then x = il + 1 by A1,A2,SCMFSA_2:71;
      hence x in {i1, il + 1} by TARSKI:def 2;
    end;
  end;
  let x be object;
   set I = a>0_goto i1;
A3: IC SCM+FSA <> a by SCMFSA_2:56;
    il in NAT by ORDINAL1:def 12;
    then
  reconsider il1 = il as Element of Values IC SCM+FSA by MEMSTR_0:def 6;
  reconsider n = il as Nat;
      reconsider u = t+*(IC SCM+FSA,il1)
       as Element of product the_Values_of SCM+FSA by CARD_3:107;
      reconsider P = Q +* (il,I) as Instruction-Sequence of SCM+FSA;
  assume
A4: x in {i1, il + 1};
  per cases by A4,TARSKI:def 2;
  suppose
A5: x = i1;
    reconsider  v = u+*(a .--> 1)
       as Element of product the_Values_of SCM+FSA by CARD_3:107;
A6: IC SCM+FSA in dom t by MEMSTR_0:2;
    not IC SCM+FSA in dom (a .--> 1) by A3,TARSKI:def 1;
    then
A8: IC v = IC u by FUNCT_4:11
      .= n by A6,FUNCT_7:31;
    il in NAT by ORDINAL1:def 12;
    then
A9:   P/.il = P.il by PBOOLE:143;
    il in NAT by ORDINAL1:def 12;
    then il in dom Q by PARTFUN1:def 2;
    then
A10: P.il = I by FUNCT_7:31;
    a in dom (a .--> 1) by TARSKI:def 1;
    then v.a = (a .--> 1).a by FUNCT_4:13
      .= 1 by FUNCOP_1:72;
    then IC Following(P,v) = i1 by A8,A10,A9,SCMFSA_2:71;
    hence thesis by A5,A8,A10,A9;
  end;
  suppose
A11: x = il + 1;
    reconsider v = u+*(a .--> 0)
      as Element of product the_Values_of SCM+FSA by CARD_3:107;
A12: IC SCM+FSA in dom t by MEMSTR_0:2;
    not IC SCM+FSA in dom (a .--> 0) by A3,TARSKI:def 1;
    then
A14: IC v = IC u by FUNCT_4:11
      .= n by A12,FUNCT_7:31;
    il in NAT by ORDINAL1:def 12;
    then
A15:   P/.il = P.il by PBOOLE:143;
    il in NAT by ORDINAL1:def 12;
    then il in dom Q by PARTFUN1:def 2;
    then
A16: P.il = I by FUNCT_7:31;
    a in dom (a .--> 0) by TARSKI:def 1;
    then v.a = (a .--> 0).a by FUNCT_4:13
      .= 0 by FUNCOP_1:72;
    then IC Following(P,v) = il + 1 by A14,A16,A15,SCMFSA_2:71;
    hence thesis by A11,A14,A16,A15;
  end;
end;
