let b, a be Int-Location ; for f being FinSeq-Location holds (product" (AddressParts (InsCode (b := f,a)))) . 2 = SCM+FSA-Data*-Loc
let f be FinSeq-Location ; (product" (AddressParts (InsCode (b := f,a)))) . 2 = SCM+FSA-Data*-Loc
dom (product" (AddressParts (InsCode (b := f,a)))) = {1,2,3}
by Th39, SCMFSA_2:50;
then A1:
2 in dom (product" (AddressParts (InsCode (b := f,a))))
by ENUMSET1:def 1;
hereby TARSKI:def 3,
XBOOLE_0:def 10 SCM+FSA-Data*-Loc c= (product" (AddressParts (InsCode (b := f,a)))) . 2
let x be
set ;
( x in (product" (AddressParts (InsCode (b := f,a)))) . 2 implies x in SCM+FSA-Data*-Loc )assume
x in (product" (AddressParts (InsCode (b := f,a)))) . 2
;
x in SCM+FSA-Data*-Loc then
x in pi (AddressParts (InsCode (b := f,a))),2
by A1, CARD_3:93;
then consider g being
Function such that A2:
g in AddressParts (InsCode (b := f,a))
and A3:
x = g . 2
by CARD_3:def 6;
consider I being
Instruction of
SCM+FSA such that A4:
g = AddressPart I
and A5:
InsCode I = InsCode (b := f,a)
by A2;
consider a,
b being
Int-Location ,
f being
FinSeq-Location such that A6:
I = b := f,
a
by A5, SCMFSA_2:50, SCMFSA_2:62;
g = <*b,f,a*>
by A4, A6, MCART_1:def 2;
then
x = f
by A3, FINSEQ_1:62;
hence
x in SCM+FSA-Data*-Loc
by SCMFSA_2:def 5;
verum
end;
let x be set ; TARSKI:def 3 ( not x in SCM+FSA-Data*-Loc or x in (product" (AddressParts (InsCode (b := f,a)))) . 2 )
assume
x in SCM+FSA-Data*-Loc
; x in (product" (AddressParts (InsCode (b := f,a)))) . 2
then reconsider x = x as FinSeq-Location by SCMFSA_2:29;
A7:
<*b,x,a*> . 2 = x
by FINSEQ_1:62;
InsCode (b := f,a) = 9
by SCMFSA_2:50;
then A8:
InsCode (b := f,a) = InsCode (b := x,a)
by SCMFSA_2:50;
AddressPart (b := x,a) = <*b,x,a*>
by MCART_1:def 2;
then
<*b,x,a*> in AddressParts (InsCode (b := f,a))
by A8;
then
x in pi (AddressParts (InsCode (b := f,a))),2
by A7, CARD_3:def 6;
hence
x in (product" (AddressParts (InsCode (b := f,a)))) . 2
by A1, CARD_3:93; verum