let il be Element of NAT ; :: thesis: SUCC il,SCM+FSA = {il,(succ il)}
set X = { ((NIC I,il) \ (JUMP I)) where I is Element of the Instructions of SCM+FSA : verum } ;
set N = {il,(succ il)};

now

let x be set ; :: thesis: ( ( x in union { ((NIC I,il) \ (JUMP I)) where I is Element of the Instructions of SCM+FSA : verum } implies x in {il,(succ il)} ) & ( x in {il,(succ il)} implies b₁ in union { ((NIC b₂,il) \ (JUMP b₂)) where I is Element of the Instructions of SCM+FSA : verum } ) )

hereby :: thesis: ( x in {il,(succ il)} implies b₁ in union { ((NIC b₂,il) \ (JUMP b₂)) where I is Element of the Instructions of SCM+FSA : verum } )

assume x in union { ((NIC I,il) \ (JUMP I)) where I is Element of the Instructions of SCM+FSA : verum } ; :: thesis: x in {il,(succ il)}
then consider Y being set such that
A1: x in Y and
A2: Y in { ((NIC I,il) \ (JUMP I)) where I is Element of the Instructions of SCM+FSA : verum } by TARSKI:def 4;
consider i being Element of the Instructions of SCM+FSA such that
A3: Y = (NIC i,il) \ (JUMP i) by A2;

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_2:120;

suppose i = [0 ,{} ] ; :: thesis: x in {il,(succ il)}

then x in {il} \ (JUMP (halt SCM+FSA )) by A1, A3, Th68, AMI_3:71, SCMFSA_2:123;
then x = il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Int-Location st i = a := b ; :: thesis: x in {il,(succ il)}

then consider a, b being Int-Location such that
A4: i = a := b ;
x in {(succ il)} \ (JUMP (a := b)) by A1, A3, A4, Th69;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Int-Location st i = AddTo a,b ; :: thesis: x in {il,(succ il)}

then consider a, b being Int-Location such that
A5: i = AddTo a,b ;
x in {(succ il)} \ (JUMP (AddTo a,b)) by A1, A3, A5, Th70;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Int-Location st i = SubFrom a,b ; :: thesis: x in {il,(succ il)}

then consider a, b being Int-Location such that
A6: i = SubFrom a,b ;
x in {(succ il)} \ (JUMP (SubFrom a,b)) by A1, A3, A6, Th71;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Int-Location st i = MultBy a,b ; :: thesis: x in {il,(succ il)}

then consider a, b being Int-Location such that
A7: i = MultBy a,b ;
x in {(succ il)} \ (JUMP (MultBy a,b)) by A1, A3, A7, Th72;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Int-Location st i = Divide a,b ; :: thesis: x in {il,(succ il)}

then consider a, b being Int-Location such that
A8: i = Divide a,b ;
x in {(succ il)} \ (JUMP (Divide a,b)) by A1, A3, A8, Th73;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex i1 being Element of NAT st i = goto i1 ; :: thesis: x in {il,(succ il)}

then consider i1 being Element of NAT such that
A9: i = goto i1 ;
x in {i1} \ (JUMP i) by A1, A3, A9, Th74;
then x in {i1} \ {i1} by A9, Th75;
hence x in {il,(succ il)} by XBOOLE_1:37; :: thesis: verum

end;

suppose ex i1 being Element of NAT ex a being Int-Location st i = a =0_goto i1 ; :: thesis: x in {il,(succ il)}

then consider i1 being Element of NAT , a being Int-Location such that
A10: i = a =0_goto i1 ;
A11: NIC i,il = {i1,(succ il)} by A10, Th76;
x in NIC i,il by A1, A3, XBOOLE_0:def 5;
then A12: ( x = i1 or x = succ il ) by A11, TARSKI:def 2;
x in (NIC i,il) \ {i1} by A1, A3, A10, Th77;
then not x in {i1} by XBOOLE_0:def 5;
hence x in {il,(succ il)} by A12, TARSKI:def 1, TARSKI:def 2; :: thesis: verum

end;

suppose ex i1 being Element of NAT ex a being Int-Location st i = a >0_goto i1 ; :: thesis: x in {il,(succ il)}

then consider i1 being Element of NAT , a being Int-Location such that
A13: i = a >0_goto i1 ;
A14: NIC i,il = {i1,(succ il)} by A13, Th78;
x in NIC i,il by A1, A3, XBOOLE_0:def 5;
then A15: ( x = i1 or x = succ il ) by A14, TARSKI:def 2;
x in (NIC i,il) \ {i1} by A1, A3, A13, Th79;
then not x in {i1} by XBOOLE_0:def 5;
hence x in {il,(succ il)} by A15, TARSKI:def 1, TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Int-Location ex f being FinSeq-Location st i = b := f,a ; :: thesis: x in {il,(succ il)}

then consider a, b being Int-Location , f being FinSeq-Location such that
A16: i = b := f,a ;
x in {(succ il)} \ (JUMP (b := f,a)) by A1, A3, A16, Th80;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Int-Location ex f being FinSeq-Location st i = f,a := b ; :: thesis: x in {il,(succ il)}

then consider a, b being Int-Location , f being FinSeq-Location such that
A17: i = f,a := b ;
x in {(succ il)} \ (JUMP (f,a := b)) by A1, A3, A17, Th81;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a being Int-Location ex f being FinSeq-Location st i = a :=len f ; :: thesis: x in {il,(succ il)}

then consider a being Int-Location , f being FinSeq-Location such that
A18: i = a :=len f ;
x in {(succ il)} \ (JUMP (a :=len f)) by A1, A3, A18, Th82;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a being Int-Location ex f being FinSeq-Location st i = f :=<0,...,0> a ; :: thesis: x in {il,(succ il)}

then consider a being Int-Location , f being FinSeq-Location such that
A19: i = f :=<0,...,0> a ;
x in {(succ il)} \ (JUMP (f :=<0,...,0> a)) by A1, A3, A19, Th83;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

end;

end;

assume A20: x in {il,(succ il)} ; :: thesis: b₁ in union { ((NIC b₂,il) \ (JUMP b₂)) where I is Element of the Instructions of SCM+FSA : verum }

per cases ( x = il or x = succ il ) by A20, TARSKI:def 2;

suppose A21: x = il ; :: thesis: b₁ in union { ((NIC b₂,il) \ (JUMP b₂)) where I is Element of the Instructions of SCM+FSA : verum }

set i = halt SCM+FSA ;
(NIC (halt SCM+FSA ),il) \ (JUMP (halt SCM+FSA )) = {il} by Th68;
then A22: {il} in { ((NIC I,il) \ (JUMP I)) where I is Element of the Instructions of SCM+FSA : verum } ;
x in {il} by A21, TARSKI:def 1;
hence x in union { ((NIC I,il) \ (JUMP I)) where I is Element of the Instructions of SCM+FSA : verum } by A22, TARSKI:def 4; :: thesis: verum

end;

suppose A23: x = succ il ; :: thesis: b₁ in union { ((NIC b₂,il) \ (JUMP b₂)) where I is Element of the Instructions of SCM+FSA : verum }

consider a, b being Int-Location ;
set i = AddTo a,b;
(NIC (AddTo a,b),il) \ (JUMP (AddTo a,b)) = {(succ il)} by Th70;
then A24: {(succ il)} in { ((NIC I,il) \ (JUMP I)) where I is Element of the Instructions of SCM+FSA : verum } ;
x in {(succ il)} by A23, TARSKI:def 1;
hence x in union { ((NIC I,il) \ (JUMP I)) where I is Element of the Instructions of SCM+FSA : verum } by A24, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence SUCC il,SCM+FSA = {il,(succ il)} by TARSKI:2; :: thesis: verum