let T be InsType of SCM ; :: thesis: ( T = 8 implies dom (product" (AddressParts T)) = {1,2} )
assume A1: T = 8 ; :: thesis: dom (product" (AddressParts T)) = {1,2}
consider i1 being Instruction-Location of SCM , a being Data-Location ;
A2: AddressPart (a >0_goto i1) = <*i1,a*> by MCART_1:def 2;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: {1,2} c= dom (product" (AddressParts T))

let x be set ; :: thesis: ( x in dom (product" (AddressParts T)) implies x in {1,2} )
assume A3: x in dom (product" (AddressParts T)) ; :: thesis: x in {1,2}
InsCode (a >0_goto i1) = 8 by MCART_1:7;
then AddressPart (a >0_goto i1) in AddressParts T by A1;
then x in dom (AddressPart (a >0_goto i1)) by A3, CARD_3:def 13;
hence x in {1,2} by A2, FINSEQ_1:4, FINSEQ_3:29; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {1,2} or x in dom (product" (AddressParts T)) )
assume A4: x in {1,2} ; :: thesis: x in dom (product" (AddressParts T))
for f being Function st f in AddressParts T holds
x in dom f

proof

let f be Function; :: thesis: ( f in AddressParts T implies x in dom f )
assume f in AddressParts T ; :: thesis: x in dom f
then consider I being Instruction of SCM such that
A5: f = AddressPart I and
A6: InsCode I = T ;
consider i1 being Instruction-Location of SCM , a being Data-Location such that
A7: I = a >0_goto i1 by A1, A6, AMI_5:54;
f = <*i1,a*> by A5, A7, MCART_1:def 2;
hence x in dom f by A4, FINSEQ_1:4, FINSEQ_3:29; :: thesis: verum

end;

hence x in dom (product" (AddressParts T)) by CARD_3:def 13; :: thesis: verum