let a, b be Int-Location ; :: thesis: (product" (AddressParts (InsCode (a := b)))) . 1 = SCM+FSA-Data-Loc
A1: InsCode (a := b) = 1 by SCMFSA_2:42;
dom (product" (AddressParts (InsCode (a := b)))) = {1,2} by Th31, SCMFSA_2:42;
then A2: 1 in dom (product" (AddressParts (InsCode (a := b)))) by TARSKI:def 2;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: SCM+FSA-Data-Loc c= (product" (AddressParts (InsCode (a := b)))) . 1

let x be set ; :: thesis: ( x in (product" (AddressParts (InsCode (a := b)))) . 1 implies x in SCM+FSA-Data-Loc )
assume x in (product" (AddressParts (InsCode (a := b)))) . 1 ; :: thesis: x in SCM+FSA-Data-Loc
then x in pi (AddressParts (InsCode (a := b))),1 by A2, CARD_3:def 13;
then consider f being Function such that
A3: f in AddressParts (InsCode (a := b)) and
A4: f . 1 = x by CARD_3:def 6;
consider I being Instruction of SCM+FSA such that
A5: f = AddressPart I and
A6: InsCode I = InsCode (a := b) by A3;
consider d1, d2 being Int-Location such that
A7: I = d1 := d2 by A6, SCMFSA_2:42, SCMFSA_2:54;
x = <*d1,d2*> . 1 by A4, A5, A7, Th18
.= d1 by FINSEQ_1:61 ;
hence x in SCM+FSA-Data-Loc by SCMFSA_2:def 4; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in SCM+FSA-Data-Loc or x in (product" (AddressParts (InsCode (a := b)))) . 1 )
assume x in SCM+FSA-Data-Loc ; :: thesis: x in (product" (AddressParts (InsCode (a := b)))) . 1
then reconsider x = x as Int-Location by SCMFSA_2:28;
InsCode (x := b) = 1 by SCMFSA_2:42;
then AddressPart (x := b) in AddressParts (InsCode (a := b)) by A1;
then A8: (AddressPart (x := b)) . 1 in pi (AddressParts (InsCode (a := b))),1 by CARD_3:def 6;
(AddressPart (x := b)) . 1 = <*x,b*> . 1 by Th18
.= x by FINSEQ_1:61 ;
hence x in (product" (AddressParts (InsCode (a := b)))) . 1 by A2, A8, CARD_3:def 13; :: thesis: verum