reserve i,j,k for Nat;

theorem Th5:
 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) = 3 holds ( Comput(P,s,k).dl.1 > 0
implies IC Comput(P,s,k+1) = 0) &
( Comput(P,s,k).dl.1 <= 0 implies IC Comput(P,s,k+1) = 4) &
  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
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) = 3;
A3: Comput(P,s,k+1)
 = Exec(P.(IC Comput(P,s,k)),Comput(P,s,k)) by EXTPRO_1:6
    .= Exec(b >0_goto 0, Comput(P,s,k)) by A1,A2,Lm3;
  hence Comput(P,s,k).b > 0 implies IC Comput(P,s,k+1) = 0 by AMI_3:9;
  thus Comput(P,s,k).b <= 0 implies IC Comput(P,s,k+1) = 4
  proof
    assume Comput(P,s,k).b <= 0;
    hence IC Comput(P,s,k+1) = IC Comput(P,s,k) + 1 by A3,AMI_3:9
      .= 4 by A2;
  end;
  thus thesis by A3,AMI_3:9;
end;
