reserve N for with_zero set;
reserve N for with_zero set;
reserve x,y,z,A,B for set,
  f,g,h for Function,
  i,j,k for Nat;
reserve S for IC-Ins-separated non empty
     with_non-empty_values AMI-Struct over N,
  s for State of S;

theorem Th3:
  for N be non empty with_zero set
  for S be IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N,
  p be NAT-defined (the InstructionsF of S)-valued Function,
  s be State of S, k be Nat holds
  Comput(p,s,k+1) = Following(p,Comput(p,s,k))
proof let N;
  let S be IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N,
  p be NAT-defined (the InstructionsF of S)-valued Function,
  s be State of S, k be Nat;
  deffunc F(set,State of S) = down Following(p,$2);
  reconsider s as Element of product the_Values_of S by CARD_3:107;
  consider f being sequence of  product the_Values_of S such that
A1: Comput(p,s,k+1) = f.(k+1) and
A2: f.0 = s and
A3: for i being Nat holds f.(i+1) = Following(p,f.i) by Def7;
  consider g being sequence of  product the_Values_of S such that
A4: Comput(p,s,k) = g.k and
A5: g.0 = s and
A6: for i being Nat holds g.(i+1) = Following(p,g.i) by Def7;
A7: for i being Nat holds f.(i+1) = F(i,f.i) by A3;
A8: for i being Nat holds g.(i+1) = F(i,g.i) by A6;
  f = g from NAT_1:sch 16(A2,A7,A5,A8);
  hence thesis by A1,A4,A6;
end;
