let i1 be Nat; :: thesis: for a being Int-Location holds (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT
let a be Int-Location; :: thesis: (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT
dom (product" (JumpParts (InsCode (a =0_goto i1)))) = {1} by Th24, SCMFSA_2:24;
then A1: 1 in dom (product" (JumpParts (InsCode (a =0_goto i1)))) by TARSKI:def 1;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: NAT c= (product" (JumpParts (InsCode (a =0_goto i1)))) . 1

let x be object ; :: thesis: ( x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 implies x in NAT )
assume x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 ; :: thesis: x in NAT
then x in pi ((JumpParts (InsCode (a =0_goto i1))),1) by A1, CARD_3:def 12;
then consider g being Function such that
A2: g in JumpParts (InsCode (a =0_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 (a =0_goto i1) by A2;
consider i2 being Nat, b being Int-Location such that
A6: I = b =0_goto i2 by A5, SCMFSA_2:24, SCMFSA_2:36;
g = <*i2*> by A4, A6, Th15;
then x = i2 by A3;
hence x in NAT by ORDINAL1:def 12; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in NAT or x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 )
assume x in NAT ; :: thesis: x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1
then reconsider x = x as Element of NAT ;
A7: <*x*> . 1 = x ;
InsCode (a =0_goto i1) = 7 by SCMFSA_2:24;
then A8: InsCode (a =0_goto i1) = InsCode (a =0_goto x) by SCMFSA_2:24;
JumpPart (a =0_goto x) = <*x*> by Th15;
then <*x*> in JumpParts (InsCode (a =0_goto i1)) by A8;
then x in pi ((JumpParts (InsCode (a =0_goto i1))),1) by A7, CARD_3:def 6;
hence x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 by A1, CARD_3:def 12; :: thesis: verum