let a, b be Int-Location ; :: thesis: for s being State of SCM+FSA holds
( (Exec (SubFrom a,b),s) . (IC SCM+FSA ) = succ (IC s) & (Exec (SubFrom a,b),s) . a = (s . a) - (s . b) & ( for c being Int-Location st c <> a holds
(Exec (SubFrom a,b),s) . c = s . c ) & ( for f being FinSeq-Location holds (Exec (SubFrom a,b),s) . f = s . f ) )
let s be State of SCM+FSA ; :: thesis: ( (Exec (SubFrom a,b),s) . (IC SCM+FSA ) = succ (IC s) & (Exec (SubFrom a,b),s) . a = (s . a) - (s . b) & ( for c being Int-Location st c <> a holds
(Exec (SubFrom a,b),s) . c = s . c ) & ( for f being FinSeq-Location holds (Exec (SubFrom a,b),s) . f = s . f ) )
consider A, B being Data-Location such that
A1: a = A and
A2: b = B and
A3: SubFrom a,b = SubFrom A,B by Def13;
reconsider S = (s | SCM-Memory ) +* (NAT --> (SubFrom A,B)) as State of SCM by Th73;
A4: Exec (SubFrom a,b),s = (s +* (Exec (SubFrom A,B),S)) +* (s | NAT ) by A3, Th75;
hence (Exec (SubFrom a,b),s) . (IC SCM+FSA ) = (Exec (SubFrom A,B),S) . (IC SCM ) by Th78
.= succ (IC S) by AMI_3:10
.= succ (IC s) by Th88 ;
:: thesis: ( (Exec (SubFrom a,b),s) . a = (s . a) - (s . b) & ( for c being Int-Location st c <> a holds
(Exec (SubFrom a,b),s) . c = s . c ) & ( for f being FinSeq-Location holds (Exec (SubFrom a,b),s) . f = s . f ) )
thus (Exec (SubFrom a,b),s) . a = (Exec (SubFrom A,B),S) . A by A1, A4, Th79
.= (S . A) - (S . B) by AMI_3:10
.= (S . A) - (s . b) by A2, Th80
.= (s . a) - (s . b) by A1, Th80 ; :: thesis: ( ( for c being Int-Location st c <> a holds
(Exec (SubFrom a,b),s) . c = s . c ) & ( for f being FinSeq-Location holds (Exec (SubFrom a,b),s) . f = s . f ) )

hereby :: thesis: for f being FinSeq-Location holds (Exec (SubFrom a,b),s) . f = s . f

let c be Int-Location ; :: thesis: ( c <> a implies (Exec (SubFrom a,b),s) . c = s . c )
assume A5: c <> a ; :: thesis: (Exec (SubFrom a,b),s) . c = s . c
reconsider C = c as Data-Location by Th25;
thus (Exec (SubFrom a,b),s) . c = (Exec (SubFrom A,B),S) . C by A4, Th79
.= S . C by A1, A5, AMI_3:10
.= s . c by Th80 ; :: thesis: verum

end;

let f be FinSeq-Location ; :: thesis: (Exec (SubFrom a,b),s) . f = s . f

A6: now

assume f in NAT ; :: thesis: contradiction
then A7: SCM+FSA-OK . f = SCM+FSA-Instr by SCMFSA_1:11;
f in SCM+FSA-Data*-Loc by Def5;
hence contradiction by A7, SCMFSA_1:12, SCMFSA_1:13; :: thesis: verum

end;

A8: not f in dom (Exec (SubFrom A,B),S) by Th68;
dom (s | NAT ) = (dom s) /\ NAT by RELAT_1:90;
then not f in dom (s | NAT ) by A6, XBOOLE_0:def 4;
hence (Exec (SubFrom a,b),s) . f = (s +* (Exec (SubFrom A,B),S)) . f by A4, FUNCT_4:12
.= s . f by A8, FUNCT_4:12 ;
:: thesis: verum