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 Initialize ((intloc 0) .--> 1) 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,k))) <> halt SCM+FSA
let s be State of SCM+FSA; :: thesis: for I being parahalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) 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,k))) <> halt SCM+FSA
let I be parahalting Program of SCM+FSA; :: thesis: ( Initialize ((intloc 0) .--> 1) 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,k))) <> halt SCM+FSA )
set s2 = s;
set P2 = P +* (loop I);
assume A2: Initialize ((intloc 0) .--> 1) 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,k))) <> halt SCM+FSA )
Start-At (0,SCM+FSA) c= Initialize ((intloc 0) .--> 1) by FUNCT_4:26;
then B3: Start-At (0,SCM+FSA) c= s by A2, XBOOLE_1:1;
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,k))) <> halt SCM+FSA
then A5: P halts_on s by B3, SCMFSA6B:def 3;
let k be Element of NAT ; :: thesis: ( k <= LifeSpan (P,s) implies CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA )
assume A6: k <= LifeSpan (P,s) ; :: thesis: CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA
set lk = IC (Comput (P,s,k));
assume A7: CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,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 A4, B3, SCMFSA6B:def 2;
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,k))) = (P +* (loop I)) . (IC (Comput ((P +* (loop I)),s,k))) by PBOOLE:158;
NPP (Comput (P,s,k)) = NPP (Comput ((P +* (loop I)),s,k)) by A5, B3, A6, Th113, A4;
then CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) = (P +* (loop I)) . (IC (Comput (P,s,k))) by A11, COMPOS_1:230
.= (loop I) . (IC (Comput (P,s,k))) by A9, A8, FUNCT_4:14 ;
hence contradiction by A7, A10, Th107; :: thesis: verum