theorem Th8: (S,U)-TruthEval m in
Funcs([:U-InterpretersOf S, S-formulasOfMaxDepth m:], BOOLEAN) &
(S,U)-TruthEval.m in
Funcs([:U-InterpretersOf S, S-formulasOfMaxDepth m:], BOOLEAN)
proof
set Fm=(S,U)-TruthEval m,II=U-InterpretersOf S,Phim=S-formulasOfMaxDepth m,
SS=AllSymbolsOf S; reconsider mm=m as Element of NAT by ORDINAL1:def 12;
reconsider Fmm=Fm as PartFunc of [:II, SS*\{{}}:], BOOLEAN;
dom Fm c= [:II,SS*\{{}}:]; then
A1: uncurry curry Fm = Fm by FUNCT_5:50;
reconsider f=curry Fm as Function of II, Funcs(Phim, BOOLEAN) by Lm17;
rng f c= Funcs(Phim, BOOLEAN) & dom f=II by FUNCT_2:def 1;
then Fm=Fm & dom Fm = [:II, Phim:] & rng Fm c= BOOLEAN by A1, FUNCT_5:26;
hence Fm in Funcs([:II, Phim:], BOOLEAN) by FUNCT_2:def 2; then
(S,U)-TruthEval.mm in Funcs([:II,Phim:],BOOLEAN) by Def20; hence thesis;
end;
