let a, b be Int-Location ; :: thesis: for s being State of SCM+FSA holds
( (Exec ((MultBy (a,b)),s)) . (IC ) = succ (IC s) & (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Int-Location st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((MultBy (a,b)),s)) . f = s . f ) )
let s be State of SCM+FSA; :: thesis: ( (Exec ((MultBy (a,b)),s)) . (IC ) = succ (IC s) & (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Int-Location st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((MultBy (a,b)),s)) . f = s . f ) )
consider A, B being Data-Location such that
A1: a = A and
A2: b = B and
A3: MultBy (a,b) = MultBy (A,B) by Def14;
reconsider S = s | SCM-Memory as State of SCM by Th73;
A4: Exec ((MultBy (a,b)),s) = s +* (Exec ((MultBy (A,B)),S)) by A3, Th75;
hence (Exec ((MultBy (a,b)),s)) . (IC ) = (Exec ((MultBy (A,B)),S)) . (IC ) by Th78
.= succ (IC S) by AMI_3:5
.= succ (IC s) by Th88 ;
:: thesis: ( (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Int-Location st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((MultBy (a,b)),s)) . f = s . f ) )
thus (Exec ((MultBy (a,b)),s)) . a = (Exec ((MultBy (A,B)),S)) . A by A1, A4, Th79
.= (S . A) * (S . B) by AMI_3:5
.= (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 ((MultBy (a,b)),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((MultBy (a,b)),s)) . f = s . f ) )

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

let c be Int-Location ; :: thesis: ( c <> a implies (Exec ((MultBy (a,b)),s)) . c = s . c )
assume A5: c <> a ; :: thesis: (Exec ((MultBy (a,b)),s)) . c = s . c
reconsider C = c as Data-Location by Th25;
thus (Exec ((MultBy (a,b)),s)) . c = (Exec ((MultBy (A,B)),S)) . C by A4, Th79
.= S . C by A1, A5, AMI_3:5
.= s . c by Th80 ; :: thesis: verum

end;

let f be FinSeq-Location ; :: thesis: (Exec ((MultBy (a,b)),s)) . f = s . f
A8: not f in dom (Exec ((MultBy (A,B)),S)) by Th68;
thus (Exec ((MultBy (a,b)),s)) . f = (s +* (Exec ((MultBy (A,B)),S))) . f by A4
.= s . f by A8, FUNCT_4:11 ; :: thesis: verum