let a, b be Int-Location ; for f being FinSeq-Location holds (product" (AddressParts (InsCode (f,a := b)))) . 1 = SCM+FSA-Data-Loc
let f be FinSeq-Location ; (product" (AddressParts (InsCode (f,a := b)))) . 1 = SCM+FSA-Data-Loc
dom (product" (AddressParts (InsCode (f,a := b)))) = {1,2,3}
by Th40, SCMFSA_2:51;
then A1:
1 in dom (product" (AddressParts (InsCode (f,a := b))))
by ENUMSET1:def 1;
hereby TARSKI:def 3,
XBOOLE_0:def 10 SCM+FSA-Data-Loc c= (product" (AddressParts (InsCode (f,a := b)))) . 1
let x be
set ;
( x in (product" (AddressParts (InsCode (f,a := b)))) . 1 implies x in SCM+FSA-Data-Loc )assume
x in (product" (AddressParts (InsCode (f,a := b)))) . 1
;
x in SCM+FSA-Data-Loc then
x in pi (AddressParts (InsCode (f,a := b))),1
by A1, CARD_3:93;
then consider g being
Function such that A2:
g in AddressParts (InsCode (f,a := b))
and A3:
x = g . 1
by CARD_3:def 6;
consider I being
Instruction of
SCM+FSA such that A4:
g = AddressPart I
and A5:
InsCode I = InsCode (f,a := b)
by A2;
consider a,
b being
Int-Location ,
f being
FinSeq-Location such that A6:
I = f,
a := b
by A5, SCMFSA_2:51, SCMFSA_2:63;
g = <*b,f,a*>
by A4, A6, MCART_1:def 2;
then
x = b
by A3, FINSEQ_1:62;
hence
x in SCM+FSA-Data-Loc
by SCMFSA_2:def 4;
verum
end;
let x be set ; TARSKI:def 3 ( not x in SCM+FSA-Data-Loc or x in (product" (AddressParts (InsCode (f,a := b)))) . 1 )
assume
x in SCM+FSA-Data-Loc
; x in (product" (AddressParts (InsCode (f,a := b)))) . 1
then reconsider x = x as Int-Location by SCMFSA_2:28;
A7:
<*x,f,a*> . 1 = x
by FINSEQ_1:62;
InsCode (f,a := b) = 10
by SCMFSA_2:51;
then A8:
InsCode (f,a := b) = InsCode (f,a := x)
by SCMFSA_2:51;
AddressPart (f,a := x) = <*x,f,a*>
by MCART_1:def 2;
then
<*x,f,a*> in AddressParts (InsCode (f,a := b))
by A8;
then
x in pi (AddressParts (InsCode (f,a := b))),1
by A7, CARD_3:def 6;
hence
x in (product" (AddressParts (InsCode (f,a := b)))) . 1
by A1, CARD_3:93; verum