reserve i,j,k for Nat;

theorem Th3:
  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) = 1 holds IC Comput(P,s,k+1) = 2 &
Comput(P,s,k+1).dl.0
  = Comput(P,s,k).dl.0 div Comput(P,s,k).dl.1 &
Comput(P,s,k+1).dl.1 =
  Comput(P,s,k).dl.0 mod Comput(P,s,k).dl.1 & Comput(P,s,k+1).dl.2 =
  Comput(P,s,k).dl.2
proof
  let s be State of SCM;
  let P be Instruction-Sequence of SCM such that
A1: Euclid-Algorithm c= P;
  let k such that
A2: IC Comput(P,s,k) = 1;
A3: Comput(P,s,k+1)
 = Exec(P.(IC Comput(P,s,k)),
Comput(P,s,k)) by EXTPRO_1:6
    .= Exec(Divide(a,b), Comput(P,s,k)) by A1,A2,Lm3;
  hence IC Comput(P,s,k+1) = IC Comput(P,s,k) + 1 by AMI_3:6
    .= 2 by A2;
  thus thesis by A3,Lm1,Lm2,AMI_3:6;
end;
