let R be Ring; :: thesis: for T being InsType of the InstructionsF of (SCM R) st T = 7 holds
dom (product" (JumpParts T)) = {1}
let T be InsType of the InstructionsF of (SCM R); :: thesis: ( T = 7 implies dom (product" (JumpParts T)) = {1} )
set i1 = the Element of NAT ;
set a = the Data-Location of R;
assume A1: T = 7 ; :: thesis: dom (product" (JumpParts T)) = {1}

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

let x be object ; :: thesis: ( x in dom (product" (JumpParts T)) implies x in {1} )
InsCode ( the Data-Location of R =0_goto the Element of NAT ) = 7 ;
then A2: JumpPart ( the Data-Location of R =0_goto the Element of NAT ) in JumpParts T by A1;
assume x in dom (product" (JumpParts T)) ; :: thesis: x in {1}
then x in DOM (JumpParts T) by CARD_3:def 12;
then x in dom (JumpPart ( the Data-Location of R =0_goto the Element of NAT )) by A2, CARD_3:108;
hence x in {1} by FINSEQ_1:2, FINSEQ_1:38; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {1} or x in dom (product" (JumpParts T)) )
assume A3: x in {1} ; :: thesis: x in dom (product" (JumpParts T))
for f being Function st f in JumpParts T holds
x in dom f

proof

let f be Function; :: thesis: ( f in JumpParts T implies x in dom f )
assume f in JumpParts T ; :: thesis: x in dom f
then consider I being Instruction of (SCM R) such that
A4: f = JumpPart I and
A5: InsCode I = T ;
consider a being Data-Location of R, i1 being Nat such that
A6: I = a =0_goto i1 by A1, A5, Th18;
f = <*i1*> by A4, A6;
hence x in dom f by A3, FINSEQ_1:2, FINSEQ_1:38; :: thesis: verum

end;

then x in DOM (JumpParts T) by CARD_3:109;
hence x in dom (product" (JumpParts T)) by CARD_3:def 12; :: thesis: verum