reserve i,j,k for Nat;

theorem Th2:
  for s being State of SCM
  for P being Instruction-Sequence of SCM
   st Euclid-Algorithm c= P
  for k st IC Comput(P,s,k) = 0 holds IC Comput(P,s,k+1) = 1 &
  Comput(P,s,k+1).dl.0 = Comput(P,s,k).dl.0 &
  Comput(P,s,k+1).dl.1 = Comput(P,s,k).dl.1 &
  Comput(P,s,k+1).dl.2 = Comput(P,s,k).dl.1
proof
  let s be State of SCM;
  let P be Instruction-Sequence of SCM such that
A1: Euclid-Algorithm c= P;
  let k;
  assume
A2: IC Comput(P,s,k) = 0;
A3: Comput(P,s,k+1)
 = Exec(P.(IC Comput(P,s,k)),Comput(P,s,k)) by EXTPRO_1:6
    .= Exec(c := b, Comput(P,s,k)) by A1,A2,Lm3;
  hence IC Comput(P,s,k+1) = IC Comput(P,s,k) + 1 by AMI_3:2
    .= 1 by A2;
  thus Comput(P,s,k+1).a = Comput(P,s,k).a & Comput(P,s,k+1).b =
  Comput(P,s,k).b by A3,AMI_3:2,10;
  thus thesis by A3,AMI_3:2;
end;
