let R be Ring; :: thesis: for i1 being Nat holds (product" (JumpParts (InsCode (goto (i1,R))))) . 1 = NAT
let i1 be Nat; :: thesis: (product" (JumpParts (InsCode (goto (i1,R))))) . 1 = NAT
dom (product" (JumpParts (InsCode (goto (i1,R))))) = {1} by Th25;
then A1: 1 in dom (product" (JumpParts (InsCode (goto (i1,R))))) by TARSKI:def 1;

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

let x be object ; :: thesis: ( x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1 implies x in NAT )
assume x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1 ; :: thesis: x in NAT
then x in pi ((JumpParts (InsCode (goto (i1,R)))),1) by A1, CARD_3:def 12;
then consider g being Function such that
A2: g in JumpParts (InsCode (goto (i1,R))) and
A3: x = g . 1 by CARD_3:def 6;
consider I being Instruction of (SCM R) such that
A4: g = JumpPart I and
A5: InsCode I = InsCode (goto (i1,R)) by A2;
consider i2 being Nat such that
A6: I = goto (i2,R) by A5, Th17;
g = <*i2*> by A4, A6;
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 (goto (i1,R))))) . 1 )
assume x in NAT ; :: thesis: x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1
then reconsider x = x as Element of NAT ;
( JumpPart (goto (x,R)) = <*x*> & InsCode (goto (i1,R)) = InsCode (goto (x,R)) ) ;
then A7: <*x*> in JumpParts (InsCode (goto (i1,R))) ;
<*x*> . 1 = x ;
then x in pi ((JumpParts (InsCode (goto (i1,R)))),1) by A7, CARD_3:def 6;
hence x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1 by A1, CARD_3:def 12; :: thesis: verum