let R be good Ring; :: thesis: for a being Data-Location of R
for i1 being Element of NAT holds (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT
let a be Data-Location of R; :: thesis: for i1 being Element of NAT holds (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT
let i1 be Element of NAT ; :: thesis: (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT
dom (product" (JumpParts (InsCode (a =0_goto i1)))) = {1} by Th15, Th39;
then A1: 1 in dom (product" (JumpParts (InsCode (a =0_goto i1)))) by TARSKI:def 1;
A2: InsCode (a =0_goto i1) = 7 by RECDEF_2: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 set ; :: 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_3:76;
then consider g being Function such that
A3: g in JumpParts (InsCode (a =0_goto i1)) and
A4: x = g . 1 by CARD_3:def 6;
consider I being Instruction of (SCM R) such that
A5: g = JumpPart I and
A6: InsCode I = InsCode (a =0_goto i1) by A3;
consider b being Data-Location of R, i2 being Element of NAT such that
A7: I = b =0_goto i2 by A2, A6, Th23;
g = <*i2*> by A5, A7, RECDEF_2:def 2;
then x = i2 by A4, FINSEQ_1:40;
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 (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 ;
( JumpPart (a =0_goto x) = <*x*> & InsCode (a =0_goto i1) = InsCode (a =0_goto x) ) by A2, RECDEF_2:def 1, RECDEF_2:def 2;
then A8: <*x*> in JumpParts (InsCode (a =0_goto i1)) ;
<*x*> . 1 = x by FINSEQ_1:40;
then x in pi ((JumpParts (InsCode (a =0_goto i1))),1) by A8, CARD_3:def 6;
hence x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 by A1, CARD_3:76; :: thesis: verum