let a be Data-Location ; :: thesis: for k1 being natural number holds (product" (AddressParts (InsCode (a >0_goto k1)))) . 2 = SCM-Data-Loc
let k1 be natural number ; :: thesis: (product" (AddressParts (InsCode (a >0_goto k1)))) . 2 = SCM-Data-Loc
dom (product" (AddressParts (InsCode (a >0_goto k1)))) = {1,2} by Th24, MCART_1:7;
then A1: 2 in dom (product" (AddressParts (InsCode (a >0_goto k1)))) by TARSKI:def 2;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: SCM-Data-Loc c= (product" (AddressParts (InsCode (a >0_goto k1)))) . 2

let x be set ; :: thesis: ( x in (product" (AddressParts (InsCode (a >0_goto k1)))) . 2 implies x in SCM-Data-Loc )
assume x in (product" (AddressParts (InsCode (a >0_goto k1)))) . 2 ; :: thesis: x in SCM-Data-Loc
then x in pi (AddressParts (InsCode (a >0_goto k1))),2 by A1, CARD_3:def 13;
then consider f being Function such that
A2: f in AddressParts (InsCode (a >0_goto k1)) and
A3: f . 2 = x by CARD_3:def 6;
consider I being Instruction of SCM such that
A4: f = AddressPart I and
A5: InsCode I = InsCode (a >0_goto k1) by A2;
consider i2 being Element of NAT , b being Data-Location such that
A6: I = b >0_goto i2 by A5, AMI_5:54, MCART_1:7;
x = <*i2,b*> . 2 by A3, A4, A6, MCART_1:def 2
.= b by FINSEQ_1:61 ;
hence x in SCM-Data-Loc by AMI_3:def 2; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in SCM-Data-Loc or x in (product" (AddressParts (InsCode (a >0_goto k1)))) . 2 )
assume x in SCM-Data-Loc ; :: thesis: x in (product" (AddressParts (InsCode (a >0_goto k1)))) . 2
then reconsider x = x as Data-Location by AMI_3:def 2;
( InsCode (a >0_goto k1) = 8 & InsCode (x >0_goto k1) = 8 ) by MCART_1:7;
then AddressPart (x >0_goto k1) in AddressParts (InsCode (a >0_goto k1)) ;
then A7: (AddressPart (x >0_goto k1)) . 2 in pi (AddressParts (InsCode (a >0_goto k1))),2 by CARD_3:def 6;
(AddressPart (x >0_goto k1)) . 2 = <*k1,x*> . 2 by MCART_1:def 2
.= x by FINSEQ_1:61 ;
hence x in (product" (AddressParts (InsCode (a >0_goto k1)))) . 2 by A1, A7, CARD_3:def 13; :: thesis: verum