let F be NAT -defined the Instructions of SCM -valued total Function; :: thesis: ( <%((dl. 0) >0_goto 1)%> ^ <%(halt SCM)%> c= F implies for i1, i2 being Integer
for s being 0 -started State-consisting of 0 ,<*i1*> ^ <*i2*> holds
( F halts_on s & LifeSpan (F,s) = 1 & ( for d being Data-Location holds (Result (F,s)) . d = s . d ) ) )
assume Z: <%((dl. 0) >0_goto 1)%> ^ <%(halt SCM)%> c= F ; :: thesis: for i1, i2 being Integer
for s being 0 -started State-consisting of 0 ,<*i1*> ^ <*i2*> holds
( F halts_on s & LifeSpan (F,s) = 1 & ( for d being Data-Location holds (Result (F,s)) . d = s . d ) )
let i1, i2 be Integer; :: thesis: for s being 0 -started State-consisting of 0 ,<*i1*> ^ <*i2*> holds
( F halts_on s & LifeSpan (F,s) = 1 & ( for d being Data-Location holds (Result (F,s)) . d = s . d ) )
let s be 0 -started State-consisting of 0 ,<*i1*> ^ <*i2*>; :: thesis: ( F halts_on s & LifeSpan (F,s) = 1 & ( for d being Data-Location holds (Result (F,s)) . d = s . d ) )
set s1 = Comput (F,s,(0 + 1));
A1: F . 0 = (dl. 0) >0_goto 1 by Z, Th15A;
A2: F . 1 = halt SCM by Z, Th15A;
A3: ( IC s = 0 & s = Comput (F,s,0) ) by COMPOS_1:def 16, EXTPRO_1:3;
s . (dl. 0) = i1 by Th15;
then A4: IC (Comput (F,s,(0 + 1))) = 0 + 1 by A1, A3, Th25;
hence F halts_on s by A2, EXTPRO_1:31; :: thesis: ( LifeSpan (F,s) = 1 & ( for d being Data-Location holds (Result (F,s)) . d = s . d ) )
thus LifeSpan (F,s) = 1 by A2, A3, A4, EXTPRO_1:34; :: thesis: for d being Data-Location holds (Result (F,s)) . d = s . d
let d be Data-Location ; :: thesis: (Result (F,s)) . d = s . d
thus (Result (F,s)) . d = (Comput (F,s,(0 + 1))) . d by A2, A4, EXTPRO_1:32
.= s . d by A1, A3, Th25 ; :: thesis: verum