let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being parahalting Program of SCM+FSA st Initialized I c= s & I c= P holds
for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA
let s be State of SCM+FSA; :: thesis: for I being parahalting Program of SCM+FSA st Initialized I c= s & I c= P holds
for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA
let I be parahalting Program of SCM+FSA; :: thesis: ( Initialized I c= s & I c= P implies for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA )
set s2 = s +* (loop I);
set P2 = P +* (loop I);
A1: ProgramPart (Initialized I) = I by SCMFSA6A:33;
assume A2: Initialized I c= s ; :: thesis: ( not I c= P or for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA )
then A3: Initialize I c= s by SCMFSA6B:8;
assume A4: I c= P ; :: thesis: for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA
then A5: P halts_on s by A2, EXTPRO_1:def 10, A1;
let k be Element of NAT ; :: thesis: ( k <= LifeSpan (P,s) implies CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA )
assume A6: k <= LifeSpan (P,s) ; :: thesis: CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA
set lk = IC (Comput (P,s,k));
assume A7: CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) = halt SCM+FSA ; :: thesis: contradiction
A8: dom I = dom (loop I) by FUNCT_4:105;
A9: IC (Comput (P,s,k)) in dom I by A3, SCMFSA6B:def 2, A4;
then A10: (loop I) . (IC (Comput (P,s,k))) in rng (loop I) by A8, FUNCT_1:def 5;
A11: (P +* (loop I)) /. (IC (Comput ((P +* (loop I)),(s +* (loop I)),k))) = (P +* (loop I)) . (IC (Comput ((P +* (loop I)),(s +* (loop I)),k))) by PBOOLE:158;
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(s +* (loop I)),k))) = (P +* (loop I)) . (IC (Comput (P,s,k))) by A5, A3, A6, Th113, A11, COMPOS_1:24, A4
.= (P +* (loop I)) . (IC (Comput (P,s,k)))
.= (loop I) . (IC (Comput (P,s,k))) by A9, A8, FUNCT_4:14 ;
hence contradiction by A7, A10, Th107; :: thesis: verum