let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
for m being Element of NAT st m < LifeSpan ((P +* I),(Initialize s)) holds
CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m)))
let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
for m being Element of NAT st m < LifeSpan ((P +* I),(Initialize s)) holds
CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m)))
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P implies for m being Element of NAT st m < LifeSpan ((P +* I),(Initialize s)) holds
CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m))) )
set s1 = Initialize s;
set P1 = P +* I;
set s2 = Initialize s;
set P2 = P +* (loop I);
A2: I c= P +* I by FUNCT_4:26;
A3: loop I c= P +* (loop I) by FUNCT_4:26;
assume that
A4: I is_closed_on s,P and
A5: I is_halting_on s,P ; :: thesis: for m being Element of NAT st m < LifeSpan ((P +* I),(Initialize s)) holds
CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m)))
let m be Element of NAT ; :: thesis: ( m < LifeSpan ((P +* I),(Initialize s)) implies CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m))) )
A6: IC (Comput ((P +* I),(Initialize s),m)) in dom I by A4, SCMFSA7B:def 7;
then A7: IC (Comput ((P +* I),(Initialize s),m)) in dom (loop I) by FUNCT_4:105;
A8: (P +* I) /. (IC (Comput ((P +* I),(Initialize s),m))) = (P +* I) . (IC (Comput ((P +* I),(Initialize s),m))) by PBOOLE:158;
A9: CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = I . (IC (Comput ((P +* I),(Initialize s),m))) by A6, A8, A2, GRFUNC_1:8;
assume A10: m < LifeSpan ((P +* I),(Initialize s)) ; :: thesis: CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m)))
A11: (P +* (loop I)) /. (IC (Comput ((P +* (loop I)),(Initialize s),m))) = (P +* (loop I)) . (IC (Comput ((P +* (loop I)),(Initialize s),m))) by PBOOLE:158;
P +* I halts_on Initialize s by A5, SCMFSA7B:def 8;
then B12: I . (IC (Comput ((P +* I),(Initialize s),m))) <> halt SCM+FSA by A10, A9, EXTPRO_1:def 14;
XX: NPP (Comput ((P +* I),(Initialize s),m)) = NPP (Comput ((P +* (loop I)),(Initialize s),m)) by A4, A5, A10, Th109;
thus CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = (P +* I) . (IC (Comput ((P +* I),(Initialize s),m))) by PBOOLE:158
.= I . (IC (Comput ((P +* I),(Initialize s),m))) by A2, A6, GRFUNC_1:8
.= (loop I) . (IC (Comput ((P +* I),(Initialize s),m))) by B12, FUNCT_4:111
.= (P +* (loop I)) . (IC (Comput ((P +* I),(Initialize s),m))) by A7, A3, GRFUNC_1:8
.= CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m))) by A11, XX, COMPOS_1:230 ; :: thesis: verum