let F be NAT -defined the InstructionsF of SCM -valued total Function; ( <%(SubFrom ((dl. 0),(dl. 1)))%> ^ <%(halt SCM)%> c= F implies for i1, i2 being Integer
for s being 0 -started State-consisting of <%i1,i2%> holds
( F halts_on s & LifeSpan (F,s) = 1 & (Result (F,s)) . (dl. 0) = i1 - i2 & ( for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d ) ) )
assume A1:
<%(SubFrom ((dl. 0),(dl. 1)))%> ^ <%(halt SCM)%> c= F
; for i1, i2 being Integer
for s being 0 -started State-consisting of <%i1,i2%> holds
( F halts_on s & LifeSpan (F,s) = 1 & (Result (F,s)) . (dl. 0) = i1 - i2 & ( for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d ) )
let i1, i2 be Integer; for s being 0 -started State-consisting of <%i1,i2%> holds
( F halts_on s & LifeSpan (F,s) = 1 & (Result (F,s)) . (dl. 0) = i1 - i2 & ( for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d ) )
let s be 0 -started State-consisting of <%i1,i2%>; ( F halts_on s & LifeSpan (F,s) = 1 & (Result (F,s)) . (dl. 0) = i1 - i2 & ( for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d ) )
set s0 = Comput (F,s,0);
set s1 = Comput (F,s,(0 + 1));
A2:
s = Comput (F,s,0)
by EXTPRO_1:2;
A3:
( s . (dl. 0) = i1 & s . (dl. 1) = i2 )
by Th2;
A4:
IC s = 0
by MEMSTR_0:def 11;
A5:
F . 0 = SubFrom ((dl. 0),(dl. 1))
by A1, Th3;
then A6:
IC (Comput (F,s,(0 + 1))) = 0 + 1
by A4, A2, Th6;
A7:
F . 1 = halt SCM
by A1, Th3;
hence
F halts_on s
by A6, EXTPRO_1:30; ( LifeSpan (F,s) = 1 & (Result (F,s)) . (dl. 0) = i1 - i2 & ( for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d ) )
thus
LifeSpan (F,s) = 1
by A4, A7, A2, A6, EXTPRO_1:33; ( (Result (F,s)) . (dl. 0) = i1 - i2 & ( for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d ) )
(Comput (F,s,(0 + 1))) . (dl. 0) = ((Comput (F,s,0)) . (dl. 0)) - ((Comput (F,s,0)) . (dl. 1))
by A4, A5, A2, Th6;
hence
(Result (F,s)) . (dl. 0) = i1 - i2
by A7, A3, A2, A6, EXTPRO_1:7; for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d
let d be Data-Location; ( d <> dl. 0 implies (Result (F,s)) . d = s . d )
assume A8:
d <> dl. 0
; (Result (F,s)) . d = s . d
thus (Result (F,s)) . d =
(Comput (F,s,(0 + 1))) . d
by A7, A6, EXTPRO_1:7
.=
s . d
by A4, A5, A2, A8, Th6
; verum