reserve k,m,n for Nat, kk,mm,nn for Element of NAT, X,Y,x,y,z for set;
reserve S,S1,S2 for Language, s,s1,s2 for Element of S;

theorem Th1: for S being Language holds
LettersOf S /\ OpSymbolsOf S ={} &
TermSymbolsOf S /\ LowerCompoundersOf S = OpSymbolsOf S &
RelSymbolsOf S \ OwnSymbolsOf S = {TheEqSymbOf S} &
OwnSymbolsOf S c= AtomicFormulaSymbolsOf S &
RelSymbolsOf S c= LowerCompoundersOf S &
OpSymbolsOf S c= TermSymbolsOf S &
LettersOf S c= TermSymbolsOf S &
TermSymbolsOf S c= OwnSymbolsOf S &
OpSymbolsOf S c= LowerCompoundersOf S &
LowerCompoundersOf S c= AtomicFormulaSymbolsOf S
proof
let S be Language;
set o=the OneF of S, z=the ZeroF of S, f=the adicity of S, R=RelSymbolsOf S,
O=OwnSymbolsOf S, SS=AllSymbolsOf S, e=TheEqSymbOf S, n=TheNorSymbOf S,
A=AtomicFormulaSymbolsOf S, D = (the carrier of S) \ {o},
L=LowerCompoundersOf S, T=TermSymbolsOf S, OP=OpSymbolsOf S,
B=LettersOf S;
thus B/\OP=f"({0}/\(NAT\{0}))
by FUNCT_1:68 .= f"({}) by XBOOLE_1:79, XBOOLE_0:def 7 .= {};
thus T/\L = f"(NAT /\ (INT \{0})) by FUNCT_1:68
.= f"((NAT/\INT)\{0}) by XBOOLE_1:49 .= OP by XBOOLE_1:28, XBOOLE_1:7;
A1: TheEqSymbOf S in R by Def17;
O =
D \ {z} by XBOOLE_1:41; then
R\O = (R\D) \/ (R /\ {z}) by XBOOLE_1:52 .=
{} \/ R /\ {z} .= {z} by ZFMISC_1:46, A1; hence R\O={TheEqSymbOf S}; thus
OwnSymbolsOf S c= AtomicFormulaSymbolsOf S by XBOOLE_1:34, ZFMISC_1:7;
f"{0} c= f"NAT & RelSymbolsOf S = f"INT \ (f"NAT) &
LowerCompoundersOf S = f"INT \ (f"{0}) by FUNCT_1:69, RELAT_1:143;
hence RelSymbolsOf S c= LowerCompoundersOf S by XBOOLE_1:34;
OpSymbolsOf S = f"NAT \ (f"{0}) by FUNCT_1:69;
hence OpSymbolsOf S c= TermSymbolsOf S;
thus LettersOf S c= TermSymbolsOf S by RELAT_1:143;
-2 = f.(TheEqSymbOf S) by Def23 .= f.z;
then not f.z in NAT; then not z in f"NAT by FUNCT_1:def 7; then
f"NAT misses {z} & f"NAT c= (( the carrier of S) \ {o}) by ZFMISC_1:50;
then f"NAT c= ((the carrier of S) \{o})\{z} by XBOOLE_1:86;
hence TermSymbolsOf S c= OwnSymbolsOf S by XBOOLE_1:41;
for x being object st x in NAT holds x in INT by INT_1:def 2;
then NAT \{0} c= INT\{0} by XBOOLE_1:33, TARSKI:def 3;
hence OpSymbolsOf S c= LowerCompoundersOf S by RELAT_1:143;
thus thesis;
end;
