let T be InsType of SCM+FSA ; :: thesis: ( T = 9 implies dom (product" (AddressParts T)) = {1,2,3} )
consider a, b being Int-Location , f being FinSeq-Location ;
assume A1: T = 9 ; :: thesis: dom (product" (AddressParts T)) = {1,2,3}
A2: AddressPart (b := f,a) = <*b,f,a*> by MCART_1:def 2;

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

let x be set ; :: thesis: ( x in dom (product" (AddressParts T)) implies x in {1,2,3} )
InsCode (b := f,a) = 9 by SCMFSA_2:50;
then A3: AddressPart (b := f,a) in AddressParts T by A1;
assume x in dom (product" (AddressParts T)) ; :: thesis: x in {1,2,3}
then x in dom (AddressPart (b := f,a)) by A3, CARD_3:def 13;
hence x in {1,2,3} by A2, FINSEQ_3:1, FINSEQ_3:30; :: thesis: verum

end;

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

proof

let g be Function; :: thesis: ( g in AddressParts T implies x in dom g )
assume g in AddressParts T ; :: thesis: x in dom g
then consider I being Instruction of SCM+FSA such that
A5: g = AddressPart I and
A6: InsCode I = T ;
consider a, b being Int-Location , f being FinSeq-Location such that
A7: I = b := f,a by A1, A6, SCMFSA_2:62;
g = <*b,f,a*> by A5, A7, MCART_1:def 2;
hence x in dom g by A4, FINSEQ_3:1, FINSEQ_3:30; :: thesis: verum

end;

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