let i1 be Element of NAT ; :: thesis: (product" (JumpParts (InsCode (goto i1)))) . 1 = NAT
dom (product" (JumpParts (InsCode (goto i1)))) = {1} by Th36, SCMFSA_2:23;
then A1: 1 in dom (product" (JumpParts (InsCode (goto i1)))) by TARSKI:def 1;

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

let x be set ; :: thesis: ( x in (product" (JumpParts (InsCode (goto i1)))) . 1 implies x in NAT )
assume x in (product" (JumpParts (InsCode (goto i1)))) . 1 ; :: thesis: x in NAT
then x in pi ((JumpParts (InsCode (goto i1))),1) by A1, CARD_3:76;
then consider g being Function such that
A2: g in JumpParts (InsCode (goto i1)) and
A3: x = g . 1 by CARD_3:def 6;
consider I being Instruction of SCM+FSA such that
A4: g = JumpPart I and
A5: InsCode I = InsCode (goto i1) by A2;
consider i2 being Element of NAT such that
A6: I = goto i2 by A5, SCMFSA_2:23, SCMFSA_2:35;
g = <*i2*> by A4, A6, RECDEF_2:def 2;
then x = i2 by A3, FINSEQ_1:def 8;
hence x in NAT ; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in NAT or x in (product" (JumpParts (InsCode (goto i1)))) . 1 )
assume x in NAT ; :: thesis: x in (product" (JumpParts (InsCode (goto i1)))) . 1
then reconsider x = x as Element of NAT ;
A7: <*x*> . 1 = x by FINSEQ_1:def 8;
InsCode (goto i1) = 6 by SCMFSA_2:23;
then A8: InsCode (goto i1) = InsCode (goto x) by SCMFSA_2:23;
JumpPart (goto x) = <*x*> by RECDEF_2:def 2;
then <*x*> in JumpParts (InsCode (goto i1)) by A8;
then x in pi ((JumpParts (InsCode (goto i1))),1) by A7, CARD_3:def 6;
hence x in (product" (JumpParts (InsCode (goto i1)))) . 1 by A1, CARD_3:76; :: thesis: verum