let l be Element of NAT ; :: thesis: for a being Int-Location
for s being State of SCM+FSA holds
( ( s . a > 0 implies (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = l ) & ( s . a <= 0 implies (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = succ (IC s) ) & ( for c being Int-Location holds (Exec ((a >0_goto l),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((a >0_goto l),s)) . f = s . f ) )
let a be Int-Location ; :: thesis: for s being State of SCM+FSA holds
( ( s . a > 0 implies (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = l ) & ( s . a <= 0 implies (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = succ (IC s) ) & ( for c being Int-Location holds (Exec ((a >0_goto l),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((a >0_goto l),s)) . f = s . f ) )
let s be State of SCM+FSA; :: thesis: ( ( s . a > 0 implies (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = l ) & ( s . a <= 0 implies (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = succ (IC s) ) & ( for c being Int-Location holds (Exec ((a >0_goto l),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((a >0_goto l),s)) . f = s . f ) )
consider A being Data-Location such that
A1: a = A and
A3: a >0_goto l = A >0_goto l by Def18;
reconsider S = (s | SCM-Memory) +* (NAT --> (A >0_goto l)) as State of SCM by Th73;
A4: Exec ((a >0_goto l),s) = (s +* (Exec ((A >0_goto l),S))) +* (s | NAT) by A3, Th75;

hereby :: thesis: ( ( s . a <= 0 implies (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = succ (IC s) ) & ( for c being Int-Location holds (Exec ((a >0_goto l),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((a >0_goto l),s)) . f = s . f ) )

assume s . a > 0 ; :: thesis: (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = l
then A5: S . A > 0 by A1, Th80;
thus (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = (Exec ((A >0_goto l),S)) . (IC SCM) by A4, Th78
.= l by A5, AMI_3:15 ; :: thesis: verum

end;

hereby :: thesis: ( ( for c being Int-Location holds (Exec ((a >0_goto l),s)) . c = s . c ) & ( for f being FinSeq-Location holds (Exec ((a >0_goto l),s)) . f = s . f ) )

assume s . a <= 0 ; :: thesis: (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = succ (IC s)
then A6: S . A <= 0 by A1, Th80;
thus (Exec ((a >0_goto l),s)) . (IC SCM+FSA) = (Exec ((A >0_goto l),S)) . (IC SCM) by A4, Th78
.= succ (IC S) by A6, AMI_3:15
.= succ (IC s) by Th88 ; :: thesis: verum

end;

hereby :: thesis: for f being FinSeq-Location holds (Exec ((a >0_goto l),s)) . f = s . f

let c be Int-Location ; :: thesis: (Exec ((a >0_goto l),s)) . c = s . c
reconsider C = c as Data-Location by Th25;
thus (Exec ((a >0_goto l),s)) . c = (Exec ((A >0_goto l),S)) . C by A4, Th79
.= S . C by AMI_3:15
.= s . c by Th80 ; :: thesis: verum

end;

let f be FinSeq-Location ; :: thesis: (Exec ((a >0_goto l),s)) . f = s . f

A7: now

assume f in NAT ; :: thesis: contradiction
then A8: SCM+FSA-OK . f = SCM+FSA-Instr by SCMFSA_1:11;
f in SCM+FSA-Data*-Loc by Def5;
hence contradiction by A8, SCMFSA_1:12, SCMFSA_1:13; :: thesis: verum

end;

A9: not f in dom (Exec ((A >0_goto l),S)) by Th68;
dom (s | NAT) = (dom s) /\ NAT by RELAT_1:90;
then not f in dom (s | NAT) by A7, XBOOLE_0:def 4;
hence (Exec ((a >0_goto l),s)) . f = (s +* (Exec ((A >0_goto l),S))) . f by A4, FUNCT_4:12
.= s . f by A9, FUNCT_4:12 ;
:: thesis: verum