let l be Instruction-Location of SCM+FSA ; :: thesis: for s being State of SCM+FSA holds
( (Exec (goto l),s) . (IC SCM+FSA ) = l & ( for c being Int-Location holds (Exec (goto l),s) . c = s . c ) & ( for f being FinSeq-Location holds (Exec (goto l),s) . f = s . f ) )
let s be State of SCM+FSA ; :: thesis: ( (Exec (goto l),s) . (IC SCM+FSA ) = l & ( for c being Int-Location holds (Exec (goto l),s) . c = s . c ) & ( for f being FinSeq-Location holds (Exec (goto l),s) . f = s . f ) )
consider La being Instruction-Location of SCM such that
A1: l = La and
A2: goto l = goto La by Def16;
reconsider S = (s | SCM-Memory ) +* (NAT --> (goto La)) as State of SCM by SCMFSA_1:18;
A3: Exec (goto l),s = (s +* (Exec (goto La),S)) +* (s | NAT ) by A2, Th75;
hence (Exec (goto l),s) . (IC SCM+FSA ) = (Exec (goto La),S) . (IC SCM ) by Th78
.= l by A1, AMI_3:13 ;
:: thesis: ( ( for c being Int-Location holds (Exec (goto l),s) . c = s . c ) & ( for f being FinSeq-Location holds (Exec (goto l),s) . f = s . f ) )

hereby :: thesis: for f being FinSeq-Location holds (Exec (goto l),s) . f = s . f

let c be Int-Location ; :: thesis: (Exec (goto l),s) . c = s . c
reconsider C = c as Data-Location by Th25;
thus (Exec (goto l),s) . c = (Exec (goto La),S) . C by A3, Th79
.= S . C by AMI_3:13
.= s . c by Th80 ; :: thesis: verum

end;

let f be FinSeq-Location ; :: thesis: (Exec (goto l),s) . f = s . f
A4: dom (s | NAT ) = (dom s) /\ NAT by RELAT_1:90;

now

assume f in NAT ; :: thesis: contradiction
then A5: SCM+FSA-OK . f = SCM+FSA-Instr by SCMFSA_1:11;
f in SCM+FSA-Data*-Loc by Def5;
hence contradiction by A5, SCMFSA_1:12, SCMFSA_1:13; :: thesis: verum

end;

then A6: not f in dom (s | NAT ) by A4, XBOOLE_0:def 4;
A7: not f in dom (Exec (goto La),S) by Th68;
thus (Exec (goto l),s) . f = (s +* (Exec (goto La),S)) . f by A3, A6, FUNCT_4:12
.= s . f by A7, FUNCT_4:12 ; :: thesis: verum