reserve a, b, d1, d2 for Data-Location,
  il, i1, i2 for Nat,
  I for Instruction of SCM,
  s, s1, s2 for State of SCM,
  T for InsType of the InstructionsF of SCM,
  k,k1 for Nat;

theorem Th19:
  NIC(a>0_goto k, il) = {k, il+1}
proof
  set t = the State of SCM,
      Q = the Instruction-Sequence of SCM;
  hereby
    let x be object;
    assume x in NIC(a>0_goto k, il);
    then consider s being Element of product the_Values_of SCM
    such that
A1: x = IC Exec(a>0_goto k,s) & IC s = il;
    per cases;
    suppose
      s.a > 0;
      then x = k by A1,AMI_3:9;
      hence x in {k, il+1} by TARSKI:def 2;
    end;
    suppose
      s.a <= 0;
      then x = il+1 by A1,AMI_3:9;
      hence x in {k, il+1} by TARSKI:def 2;
    end;
  end;
  let x be object;
  set I = a>0_goto k;
A2: IC SCM <> a by AMI_5:2;
  assume
A3: x in {k,il+1};
   reconsider n = il as Element of NAT by ORDINAL1:def 12;
  reconsider il1 = n as Element of Values IC SCM by MEMSTR_0:def 6;
      reconsider u = t+*(IC SCM,il1)
       as Element of product the_Values_of SCM by CARD_3:107;
      reconsider P = Q +* (il,I)
       as Instruction-Sequence of SCM;
  per cases by A3,TARSKI:def 2;
  suppose
A4: x = k;
    reconsider v = u+*(a .--> 1)
     as Element of product the_Values_of SCM by CARD_3:107;
A5: IC SCM in dom t by MEMSTR_0:2;
    not IC SCM in dom (a .--> 1) by A2,TARSKI:def 1;
    then
A7: IC v = IC u by FUNCT_4:11
      .= n by A5,FUNCT_7:31;
     reconsider il as Element of NAT by ORDINAL1:def 12;
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 .--> 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) = k by A7,A9,A8,AMI_3:9;
    hence thesis by A4,A7,A9,A8;
  end;
  suppose
A10: x = il+1;
    reconsider v = u+*(a .--> 0)
     as Element of product the_Values_of SCM by CARD_3:107;
A11: IC SCM in dom t by MEMSTR_0:2;
    not IC SCM in dom (a .--> 0) by A2,TARSKI:def 1;
    then
A13: IC v = IC u by FUNCT_4:11
      .= n by A11,FUNCT_7:31;
     reconsider il as Element of NAT by ORDINAL1:def 12;
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 .--> 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 A13,A15,A14,AMI_3:9;
    hence thesis by A10,A13,A15,A14;
  end;
end;
