let F be NAT -defined the Instructions of SCM -valued total Function; :: thesis: ( <%(MultBy ((dl. 0),(dl. 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 & (Result (F,s)) . (dl. 0) = i1 * i2 & ( for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d ) ) )
assume Z: <%(MultBy ((dl. 0),(dl. 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 & (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; :: thesis: for s being 0 -started State-consisting of 0 ,<*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 0 ,<*i1*> ^ <*i2*>; :: thesis: ( 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));
A1: s = Comput (F,s,0) by EXTPRO_1:3;
A2: ( s . (dl. 0) = i1 & s . (dl. 1) = i2 ) by Th15;
A3: IC s = 0 by COMPOS_1:def 16;
A4: F . 0 = MultBy ((dl. 0),(dl. 1)) by Z, Th15A;
then A5: IC (Comput (F,s,(0 + 1))) = 0 + 1 by A3, A1, Th21;
A6: F . 1 = halt SCM by Z, Th15A;
hence F halts_on s by A5, EXTPRO_1:31; :: thesis: ( 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 A3, A6, A1, A5, EXTPRO_1:34; :: thesis: ( (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 A3, A4, A1, Th21;
hence (Result (F,s)) . (dl. 0) = i1 * i2 by A6, A2, A1, A5, EXTPRO_1:32; :: thesis: for d being Data-Location st d <> dl. 0 holds
(Result (F,s)) . d = s . d
let d be Data-Location ; :: thesis: ( d <> dl. 0 implies (Result (F,s)) . d = s . d )
assume A7: d <> dl. 0 ; :: thesis: (Result (F,s)) . d = s . d
thus (Result (F,s)) . d = (Comput (F,s,(0 + 1))) . d by A6, A5, EXTPRO_1:32
.= s . d by A3, A4, A1, A7, Th21 ; :: thesis: verum