let E be non empty set ; :: thesis: for A being non empty automaton over (Lex E) \/ {(<%> E)} holds Lang A = Lang (_bool A)
let A be non empty automaton over (Lex E) \/ {(<%> E)}; :: thesis: Lang A = Lang (_bool A)
set DA = _bool A;
A1: for w being Element of E ^omega st w in Lang A holds
w in Lang (_bool A)

proof

let w be Element of E ^omega ; :: thesis: ( w in Lang A implies w in Lang (_bool A) )
assume w in Lang A ; :: thesis: w in Lang (_bool A)
then w -succ_of ( the InitS of A,A) meets the FinalS of A by Th19;
then ex x being object st
( x in w -succ_of ( the InitS of A,A) & x in the FinalS of A ) by XBOOLE_0:3;
then A2: w -succ_of ( the InitS of A,A) in the FinalS of (_bool A) by Th33;
w -succ_of ( the InitS of (_bool A),(_bool A)) = {(w -succ_of ( the InitS of A,A))} by Th37;
then w -succ_of ( the InitS of A,A) in w -succ_of ( the InitS of (_bool A),(_bool A)) by TARSKI:def 1;
then w -succ_of ( the InitS of (_bool A),(_bool A)) meets the FinalS of (_bool A) by A2, XBOOLE_0:3;
hence w in Lang (_bool A) by Th19; :: thesis: verum

end;

for w being Element of E ^omega st w in Lang (_bool A) holds
w in Lang A

proof

let w be Element of E ^omega ; :: thesis: ( w in Lang (_bool A) implies w in Lang A )
assume w in Lang (_bool A) ; :: thesis: w in Lang A
then w -succ_of ( the InitS of (_bool A),(_bool A)) meets the FinalS of (_bool A) by Th19;
then consider x being object such that
A3: x in w -succ_of ( the InitS of (_bool A),(_bool A)) and
A4: x in the FinalS of (_bool A) by XBOOLE_0:3;
w -succ_of ( the InitS of (_bool A),(_bool A)) = {(w -succ_of ( the InitS of A,A))} by Th37;
then x = w -succ_of ( the InitS of A,A) by A3, TARSKI:def 1;
then w -succ_of ( the InitS of A,A) meets the FinalS of A by A4, Th34;
hence w in Lang A by Th19; :: thesis: verum

end;

hence Lang A = Lang (_bool A) by A1, SUBSET_1:3; :: thesis: verum