let I be Instruction of SCM+FSA; :: according to AMISTD_2:def 2 :: thesis: I is with_explicit_jumps
thus JUMP I c= rng (JumpPart I) :: according to AMISTD_2:def 1,XBOOLE_0:def 10 :: thesis: proj2 (I `2_3) c= JUMP I

proof

let f be set ; :: according to TARSKI:def 3 :: thesis: ( not f in JUMP I or f in rng (JumpPart I) )
assume A1: f in JUMP I ; :: thesis: f in rng (JumpPart I)

per cases ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex i1 being Element of NAT st I = goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 or ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) or ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) by SCMFSA_2:93;

suppose I = [0,{},{}] ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1, SCMFSA_2:96; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = a := b ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = AddTo (a,b) ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = SubFrom (a,b) ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = MultBy (a,b) ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = Divide (a,b) ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

suppose A2: ex i1 being Element of NAT st I = goto i1 ; :: thesis: f in rng (JumpPart I)

consider i1 being Element of NAT such that
A3: I = goto i1 by A2;
A4: JumpPart (goto i1) = <*i1*> by RECDEF_2:def 2;
rng <*i1*> = {i1} by FINSEQ_1:39;
hence f in rng (JumpPart I) by A1, A3, A4, Th34; :: thesis: verum

end;

suppose A5: ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 ; :: thesis: f in rng (JumpPart I)

consider a being Int-Location, i1 being Element of NAT such that
A6: I = a =0_goto i1 by A5;
A7: JumpPart (a =0_goto i1) = <*i1*> by Th15;
rng <*i1*> = {i1} by FINSEQ_1:39;
hence f in rng (JumpPart I) by A1, A6, A7, Th36; :: thesis: verum

end;

suppose A8: ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 ; :: thesis: f in rng (JumpPart I)

consider a being Int-Location, i1 being Element of NAT such that
A9: I = a >0_goto i1 by A8;
A10: JumpPart (a >0_goto i1) = <*i1*> by Th16;
rng <*i1*> = {i1} by FINSEQ_1:39;
hence f in rng (JumpPart I) by A1, A9, A10, Th38; :: thesis: verum

end;

suppose ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

suppose ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

suppose ex a being Int-Location ex f being FinSeq-Location st I = a :=len f ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

suppose ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ; :: thesis: f in rng (JumpPart I)

hence f in rng (JumpPart I) by A1; :: thesis: verum

end;

end;

end;

let f be set ; :: according to TARSKI:def 3 :: thesis: ( not f in proj2 (I `2_3) or f in JUMP I )
assume f in rng (JumpPart I) ; :: thesis: f in JUMP I
then consider k being set such that
A11: k in dom (JumpPart I) and
A12: f = (JumpPart I) . k by FUNCT_1:def 3;

per cases ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex i1 being Element of NAT st I = goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 or ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) or ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) by SCMFSA_2:93;

suppose I = [0,{},{}] ; :: thesis: f in JUMP I

then dom (JumpPart I) = dom {} by RECDEF_2:def 2;
hence f in JUMP I by A11; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = a := b ; :: thesis: f in JUMP I

then consider a, b being Int-Location such that
A13: I = a := b ;
k in dom {} by A11, A13, Th10;
hence f in JUMP I ; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = AddTo (a,b) ; :: thesis: f in JUMP I

then consider a, b being Int-Location such that
A14: I = AddTo (a,b) ;
k in dom {} by A11, A14, Th11;
hence f in JUMP I ; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = SubFrom (a,b) ; :: thesis: f in JUMP I

then consider a, b being Int-Location such that
A15: I = SubFrom (a,b) ;
k in dom {} by A11, A15, Th12;
hence f in JUMP I ; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = MultBy (a,b) ; :: thesis: f in JUMP I

then consider a, b being Int-Location such that
A16: I = MultBy (a,b) ;
k in dom {} by A11, A16, Th13;
hence f in JUMP I ; :: thesis: verum

end;

suppose ex a, b being Int-Location st I = Divide (a,b) ; :: thesis: f in JUMP I

then consider a, b being Int-Location such that
A17: I = Divide (a,b) ;
k in dom {} by A11, A17, Th14;
hence f in JUMP I ; :: thesis: verum

end;

suppose ex i1 being Element of NAT st I = goto i1 ; :: thesis: f in JUMP I

then consider i1 being Element of NAT such that
A18: I = goto i1 ;
A19: JumpPart I = <*i1*> by A18, RECDEF_2:def 2;
then k = 1 by A11, FINSEQ_1:90;
then A20: f = i1 by A19, A12, FINSEQ_1:def 8;
JUMP I = {i1} by A18, Th34;
hence f in JUMP I by A20, TARSKI:def 1; :: thesis: verum

end;

suppose ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 ; :: thesis: f in JUMP I

then consider a being Int-Location, i1 being Element of NAT such that
A21: I = a =0_goto i1 ;
A22: JumpPart I = <*i1*> by A21, Th15;
then k = 1 by A11, FINSEQ_1:90;
then A23: f = i1 by A22, A12, FINSEQ_1:def 8;
JUMP I = {i1} by A21, Th36;
hence f in JUMP I by A23, TARSKI:def 1; :: thesis: verum

end;

suppose ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 ; :: thesis: f in JUMP I

then consider a being Int-Location, i1 being Element of NAT such that
A24: I = a >0_goto i1 ;
A25: JumpPart I = <*i1*> by A24, Th16;
then k = 1 by A11, FINSEQ_1:90;
then A26: f = i1 by A25, A12, FINSEQ_1:def 8;
JUMP I = {i1} by A24, Th38;
hence f in JUMP I by A26, TARSKI:def 1; :: thesis: verum

end;

suppose ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) ; :: thesis: f in JUMP I

then consider a, b being Int-Location, f being FinSeq-Location such that
A27: I = b := (f,a) ;
k in dom {} by A11, A27, RECDEF_2:def 2;
hence f in JUMP I ; :: thesis: verum

end;

suppose ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b ; :: thesis: f in JUMP I

then consider a, b being Int-Location, f being FinSeq-Location such that
A28: I = (f,a) := b ;
k in dom {} by A11, A28, RECDEF_2:def 2;
hence f in JUMP I ; :: thesis: verum

end;

suppose ex a being Int-Location ex f being FinSeq-Location st I = a :=len f ; :: thesis: f in JUMP I

then consider a being Int-Location, f being FinSeq-Location such that
A29: I = a :=len f ;
k in dom {} by A11, A29, RECDEF_2:def 2;
hence f in JUMP I ; :: thesis: verum

end;

suppose ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ; :: thesis: f in JUMP I

then consider a being Int-Location, f being FinSeq-Location such that
A30: I = f :=<0,...,0> a ;
k in dom {} by A11, A30, RECDEF_2:def 2;
hence f in JUMP I ; :: thesis: verum

end;

end;