set f = the_Values_of SCM+FSA;
set s = the SCM+FSA-State;
assume A1:
goto la is halting
; contradiction
reconsider a3 = la as Nat ;
set t = the SCM+FSA-State +* (NAT .--> (a3 + 1));
NAT in dom (NAT .--> (a3 + 1))
by TARSKI:def 1;
then A2: ( the SCM+FSA-State +* (NAT .--> (a3 + 1))) . NAT =
(NAT .--> (a3 + 1)) . NAT
by FUNCT_4:13
.=
a3 + 1
by FUNCOP_1:72
;
A3:
for x being object st x in dom (the_Values_of SCM+FSA) holds
( the SCM+FSA-State +* (NAT .--> (a3 + 1))) . x in (the_Values_of SCM+FSA) . x
A5:
{NAT} c= SCM+FSA-Memory
by SCMFSA_1:5, ZFMISC_1:31;
A6: dom ( the SCM+FSA-State +* (NAT .--> (a3 + 1))) =
(dom the SCM+FSA-State) \/ (dom (NAT .--> (a3 + 1)))
by FUNCT_4:def 1
.=
SCM+FSA-Memory \/ (dom (NAT .--> (a3 + 1)))
by SCMFSA_1:33
.=
SCM+FSA-Memory \/ {NAT}
.=
SCM+FSA-Memory
by A5, XBOOLE_1:12
;
dom (the_Values_of SCM+FSA) = SCM+FSA-Memory
by SCMFSA_1:32;
then reconsider t = the SCM+FSA-State +* (NAT .--> (a3 + 1)) as State of SCM+FSA by A6, A3, FUNCT_1:def 14, PARTFUN1:def 2, RELAT_1:def 18;
reconsider w = t as SCM+FSA-State by CARD_3:107;
NAT in dom (NAT .--> la)
by TARSKI:def 1;
then A7: (w +* (NAT .--> la)) . NAT =
(NAT .--> la) . NAT
by FUNCT_4:13
.=
la
by FUNCOP_1:72
;
(w +* (NAT .--> la)) . NAT =
(SCM+FSA-Chg (w,a3)) . NAT
.=
a3
by SCMFSA_1:19
.=
(Exec ((goto la),t)) . NAT
by Th1, Th62
.=
t . NAT
by A1
;
hence
contradiction
by A2, A7; verum