let i1 be Instruction-Location of SCM+FSA ; :: thesis: (product" (AddressParts (InsCode (goto i1)))) . 1 = NAT
A1: InsCode (goto i1) = 6 by SCMFSA_2:47;
dom (product" (AddressParts (InsCode (goto i1)))) = {1} by Th36, SCMFSA_2:47;
then A2: 1 in dom (product" (AddressParts (InsCode (goto i1)))) by TARSKI:def 1;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: NAT c= (product" (AddressParts (InsCode (goto i1)))) . 1

let x be set ; :: thesis: ( x in (product" (AddressParts (InsCode (goto i1)))) . 1 implies x in NAT )
assume x in (product" (AddressParts (InsCode (goto i1)))) . 1 ; :: thesis: x in NAT
then x in pi (AddressParts (InsCode (goto i1))),1 by A2, CARD_3:93;
then consider g being Function such that
A3: g in AddressParts (InsCode (goto i1)) and
A4: x = g . 1 by CARD_3:def 6;
consider I being Instruction of SCM+FSA such that
A5: g = AddressPart I and
A6: InsCode I = InsCode (goto i1) by A3;
consider i2 being Instruction-Location of SCM+FSA such that
A7: I = goto i2 by A1, A6, SCMFSA_2:59;
g = <*i2*> by A5, A7, Th23;
then x = i2 by A4, FINSEQ_1:def 8;
hence x in NAT by AMI_1:def 4; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in NAT or x in (product" (AddressParts (InsCode (goto i1)))) . 1 )
assume x in NAT ; :: thesis: x in (product" (AddressParts (InsCode (goto i1)))) . 1
then reconsider x = x as Instruction-Location of SCM+FSA by AMI_1:def 4;
A8: AddressPart (goto x) = <*x*> by Th23;
InsCode (goto i1) = InsCode (goto x) by A1, SCMFSA_2:47;
then A9: <*x*> in AddressParts (InsCode (goto i1)) by A8;
<*x*> . 1 = x by FINSEQ_1:def 8;
then x in pi (AddressParts (InsCode (goto i1))),1 by A9, CARD_3:def 6;
hence x in (product" (AddressParts (InsCode (goto i1)))) . 1 by A2, CARD_3:93; :: thesis: verum