let k be Element of NAT ; :: thesis: for s being 0 -started State of SCM
for P being the Instructions of SCM -valued ManySortedSet of NAT st Euclide-Algorithm c= P & s . (dl. 0 ) > 0 & s . (dl. 1) > 0 holds
( (Comput P,s,(4 * k)) . (dl. 0 ) > 0 & ( ( (Comput P,s,(4 * k)) . (dl. 1) > 0 & IC (Comput P,s,(4 * k)) = 0 ) or ( (Comput P,s,(4 * k)) . (dl. 1) = 0 & IC (Comput P,s,(4 * k)) = 4 ) ) )
let s be 0 -started State of SCM ; :: thesis: for P being the Instructions of SCM -valued ManySortedSet of NAT st Euclide-Algorithm c= P & s . (dl. 0 ) > 0 & s . (dl. 1) > 0 holds
( (Comput P,s,(4 * k)) . (dl. 0 ) > 0 & ( ( (Comput P,s,(4 * k)) . (dl. 1) > 0 & IC (Comput P,s,(4 * k)) = 0 ) or ( (Comput P,s,(4 * k)) . (dl. 1) = 0 & IC (Comput P,s,(4 * k)) = 4 ) ) )
let P be the Instructions of SCM -valued ManySortedSet of NAT ; :: thesis: ( Euclide-Algorithm c= P & s . (dl. 0 ) > 0 & s . (dl. 1) > 0 implies ( (Comput P,s,(4 * k)) . (dl. 0 ) > 0 & ( ( (Comput P,s,(4 * k)) . (dl. 1) > 0 & IC (Comput P,s,(4 * k)) = 0 ) or ( (Comput P,s,(4 * k)) . (dl. 1) = 0 & IC (Comput P,s,(4 * k)) = 4 ) ) ) )
assume that
A2: Euclide-Algorithm c= P and
A3: ( s . (dl. 0 ) > 0 & s . (dl. 1) > 0 ) ; :: thesis: ( (Comput P,s,(4 * k)) . (dl. 0 ) > 0 & ( ( (Comput P,s,(4 * k)) . (dl. 1) > 0 & IC (Comput P,s,(4 * k)) = 0 ) or ( (Comput P,s,(4 * k)) . (dl. 1) = 0 & IC (Comput P,s,(4 * k)) = 4 ) ) )
A1: IC s = 0 by COMPOS_1:def 20;
defpred S₂[ Element of NAT ] means ( (Comput P,s,(4 * $1)) . (dl. 0 ) > 0 & ( ( (Comput P,s,(4 * $1)) . (dl. 1) > 0 & IC (Comput P,s,(4 * $1)) = 0 ) or ( (Comput P,s,(4 * $1)) . (dl. 1) = 0 & IC (Comput P,s,(4 * $1)) = 4 ) ) );
A4: for k being Element of NAT st S₂[k] holds
S₂[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₂[k] implies S₂[k + 1] )
set c4 = Comput P,s,(4 * k);
set c5 = Comput P,s,((4 * k) + 1);
set c6 = Comput P,s,((4 * k) + 2);
set c7 = Comput P,s,((4 * k) + 3);
set c8 = Comput P,s,((4 * k) + 4);
A5: Comput P,s,((4 * k) + 3) = Comput P,s,(((4 * k) + 2) + 1) ;
A6: Comput P,s,((4 * k) + 4) = Comput P,s,(((4 * k) + 3) + 1) ;
assume A7: (Comput P,s,(4 * k)) . (dl. 0 ) > 0 ; :: thesis: ( ( not ( (Comput P,s,(4 * k)) . (dl. 1) > 0 & IC (Comput P,s,(4 * k)) = 0 ) & not ( (Comput P,s,(4 * k)) . (dl. 1) = 0 & IC (Comput P,s,(4 * k)) = 4 ) ) or S₂[k + 1] )
assume A8: ( ( (Comput P,s,(4 * k)) . (dl. 1) > 0 & IC (Comput P,s,(4 * k)) = 0 ) or ( (Comput P,s,(4 * k)) . (dl. 1) = 0 & IC (Comput P,s,(4 * k)) = 4 ) ) ; :: thesis: S₂[k + 1]
A9: Comput P,s,((4 * k) + 2) = Comput P,s,(((4 * k) + 1) + 1) ;

now

per cases ( (Comput P,s,(4 * k)) . (dl. 1) > 0 or (Comput P,s,(4 * k)) . (dl. 1) = 0 ) by A8;

case A10: (Comput P,s,(4 * k)) . (dl. 1) > 0 ; :: thesis: ( (Comput P,s,((4 * k) + 4)) . (dl. 0 ) > 0 & ( ( (Comput P,s,((4 * k) + 4)) . (dl. 1) > 0 & IC (Comput P,s,((4 * k) + 4)) = 0 ) or ( (Comput P,s,((4 * k) + 4)) . (dl. 1) = 0 & IC (Comput P,s,((4 * k) + 4)) = 4 ) ) )

then A11: IC (Comput P,s,((4 * k) + 1)) = 1 by A2, A8, Th5;
then A12: IC (Comput P,s,((4 * k) + 2)) = 2 by A2, A9, Th6;
then A13: IC (Comput P,s,((4 * k) + 3)) = 3 by A2, A5, Th7;
then A14: (Comput P,s,((4 * k) + 4)) . (dl. 1) = (Comput P,s,((4 * k) + 3)) . (dl. 1) by A2, A6, Th8;
A15: ( (Comput P,s,((4 * k) + 3)) . (dl. 0 ) = (Comput P,s,((4 * k) + 2)) . (dl. 2) & (Comput P,s,((4 * k) + 3)) . (dl. 1) = (Comput P,s,((4 * k) + 2)) . (dl. 1) ) by A2, A5, A12, Th7;
A16: ( (Comput P,s,((4 * k) + 2)) . (dl. 1) = ((Comput P,s,((4 * k) + 1)) . (dl. 0 )) mod ((Comput P,s,((4 * k) + 1)) . (dl. 1)) & (Comput P,s,((4 * k) + 2)) . (dl. 2) = (Comput P,s,((4 * k) + 1)) . (dl. 2) ) by A2, A9, A11, Th6;
A17: ( (Comput P,s,((4 * k) + 1)) . (dl. 1) = (Comput P,s,(4 * k)) . (dl. 1) & (Comput P,s,((4 * k) + 1)) . (dl. 2) = (Comput P,s,(4 * k)) . (dl. 1) ) by A2, A8, A10, Th5;
( (Comput P,s,((4 * k) + 3)) . (dl. 1) > 0 implies IC (Comput P,s,((4 * k) + 4)) = 0 ) by A2, A6, A13, Th8;
hence ( (Comput P,s,((4 * k) + 4)) . (dl. 0 ) > 0 & ( ( (Comput P,s,((4 * k) + 4)) . (dl. 1) > 0 & IC (Comput P,s,((4 * k) + 4)) = 0 ) or ( (Comput P,s,((4 * k) + 4)) . (dl. 1) = 0 & IC (Comput P,s,((4 * k) + 4)) = 4 ) ) ) by A2, A6, A10, A17, A16, A13, A15, A14, Th8, NEWTON:78; :: thesis: verum

end;

case (Comput P,s,(4 * k)) . (dl. 1) = 0 ; :: thesis: ( (Comput P,s,((4 * k) + 4)) . (dl. 0 ) > 0 & (Comput P,s,((4 * k) + 4)) . (dl. 1) = 0 & IC (Comput P,s,((4 * k) + 4)) = 4 )

hence ( (Comput P,s,((4 * k) + 4)) . (dl. 0 ) > 0 & (Comput P,s,((4 * k) + 4)) . (dl. 1) = 0 & IC (Comput P,s,((4 * k) + 4)) = 4 ) by A2, A7, A8, Th9; :: thesis: verum

end;

end;

end;

hence S₂[k + 1] ; :: thesis: verum

end;

A18: S₂[ 0 ] by A1, A3, AMI_1:13;
for k being Element of NAT holds S₂[k] from NAT_1:sch 1(A18, A4);
hence ( (Comput P,s,(4 * k)) . (dl. 0 ) > 0 & ( ( (Comput P,s,(4 * k)) . (dl. 1) > 0 & IC (Comput P,s,(4 * k)) = 0 ) or ( (Comput P,s,(4 * k)) . (dl. 1) = 0 & IC (Comput P,s,(4 * k)) = 4 ) ) ) ; :: thesis: verum