let T be InsType of the InstructionsF of SCM+FSA; :: thesis: ( T = 7 implies dom (product" (JumpParts T)) = {1} )
set i1 = the Nat;
set a = the Int-Location;
assume A1: T = 7 ; :: thesis: dom (product" (JumpParts T)) = {1}
A2: JumpPart ( the Int-Location =0_goto the Nat) = <* the Nat*> by Th15;

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 Int-Location =0_goto the Nat) = 7 by SCMFSA_2:24;
then A3: JumpPart ( the Int-Location =0_goto the 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 Int-Location =0_goto the Nat)) by A3, CARD_3:108;
hence x in {1} by A2, 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 A4: 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+FSA such that
A5: f = JumpPart I and
A6: InsCode I = T ;
consider i1 being Nat, a being Int-Location such that
A7: I = a =0_goto i1 by A1, A6, SCMFSA_2:36;
f = <*i1*> by A5, A7, Th15;
hence x in dom f by A4, 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