reserve x for set,
  D for non empty set,
  k, n for Nat,
  z for Nat;
reserve
  N for with_zero set,
  S for IC-Ins-separated non empty
          with_non-empty_values AMI-Struct over N,
  i for Element of the InstructionsF of S,
  l, l1, l2, l3 for Nat,
  s for State of S;
reserve ss for Element of product the_Values_of S;
reserve T for standard IC-Ins-separated non empty
     with_non-empty_values AMI-Struct over N;

theorem
 for i being Instruction of STC N, s being State of STC N st InsCode i = 1
  holds Exec(i,s) = IncIC(s,1)
proof
  let i be Instruction of STC N, s be State of STC N;
  set M = STC N;
  assume
A1: InsCode i = 1;
A2: now
    assume i in {[0,0,0]};
    then i = [0,0,0] by TARSKI:def 1;
    hence contradiction by A1;
  end;
  the InstructionsF of M = III by Def7;
  then i = [1,0,0] or i = [0,0,0] by TARSKI:def 2;
  then
A3: i in {[1,0,0]} by A1,TARSKI:def 1;
  consider f be Function of product the_Values_of M, product
  the_Values_of M such that
A4: for s being Element of product the_Values_of M
   holds f.s = s+*(0 .-->(In(s.0,NAT)+1)) and
A5: the Execution of M = ([1,0,0] .--> f) +* ([0,0,0] .--> id product
  the_Values_of M) by Def7;
A6: for s being State of M holds f.s = s+*(0 .-->(In(s.0,NAT)+1))
 proof let s be State of M;
  reconsider s as Element of product the_Values_of M by CARD_3:107;
  f.s = s+*(0 .-->(In(s.0,NAT)+1)) by A4;
  hence thesis;
 end;
A7: IC M = 0 by Def7;
A8: s in product the_Values_of M by CARD_3:107;
   dom(the_Values_of M) = the carrier of M by PARTFUN1:def 2
     .= {0} by Def7;
    then
A9:  0 in dom(the_Values_of M) by TARSKI:def 1;
  Values IC M = NAT by MEMSTR_0:def 6;
  then reconsider k = s.0 as Element of NAT by A8,A9,CARD_3:9,A7;
A10: Start-At(IC s+1,M) = 0 .-->(In(k,NAT)+1) by A7;
  dom ([0,0,0] .--> id product the_Values_of M) = {[0,0,0]};
  then (the Execution of M).i = ([1,0,0] .--> f).i by A5,A2,FUNCT_4:11
    .= f by A3,FUNCOP_1:7;
  hence Exec(i,s) = IncIC(s,1) by A10,A6;
end;
