let s be State of SCM; :: thesis: for P being Instruction-Sequence of SCM st Euclid-Algorithm c= P holds
for k being Nat st IC (Comput (P,s,k)) = 1 holds
( IC (Comput (P,s,(k + 1))) = 2 & (Comput (P,s,(k + 1))) . () = ((Comput (P,s,k)) . ()) div ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 1) = ((Comput (P,s,k)) . ()) mod ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) )

let P be Instruction-Sequence of SCM; :: thesis: ( Euclid-Algorithm c= P implies for k being Nat st IC (Comput (P,s,k)) = 1 holds
( IC (Comput (P,s,(k + 1))) = 2 & (Comput (P,s,(k + 1))) . () = ((Comput (P,s,k)) . ()) div ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 1) = ((Comput (P,s,k)) . ()) mod ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) ) )

assume A1: Euclid-Algorithm c= P ; :: thesis: for k being Nat st IC (Comput (P,s,k)) = 1 holds
( IC (Comput (P,s,(k + 1))) = 2 & (Comput (P,s,(k + 1))) . () = ((Comput (P,s,k)) . ()) div ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 1) = ((Comput (P,s,k)) . ()) mod ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) )

let k be Nat; :: thesis: ( IC (Comput (P,s,k)) = 1 implies ( IC (Comput (P,s,(k + 1))) = 2 & (Comput (P,s,(k + 1))) . () = ((Comput (P,s,k)) . ()) div ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 1) = ((Comput (P,s,k)) . ()) mod ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) ) )
assume A2: IC (Comput (P,s,k)) = 1 ; :: thesis: ( IC (Comput (P,s,(k + 1))) = 2 & (Comput (P,s,(k + 1))) . () = ((Comput (P,s,k)) . ()) div ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 1) = ((Comput (P,s,k)) . ()) mod ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) )
A3: Comput (P,s,(k + 1)) = Exec ((P . (IC (Comput (P,s,k)))),(Comput (P,s,k))) by EXTPRO_1:6
.= Exec ((Divide ((),(dl. 1))),(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 ;
:: thesis: ( (Comput (P,s,(k + 1))) . () = ((Comput (P,s,k)) . ()) div ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 1) = ((Comput (P,s,k)) . ()) mod ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) )
thus ( (Comput (P,s,(k + 1))) . () = ((Comput (P,s,k)) . ()) div ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 1) = ((Comput (P,s,k)) . ()) mod ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) ) by ; :: thesis: verum