theorem for T being |.ar a.|-element Element of (AllTermsOf S)* holds
(a is not relational implies (X-freeInterpreter(a)).T = a-compound T) &
(a is relational implies
(X-freeInterpreter(a)).T = (chi(X,AtomicFormulasOf S)).(a-compound T))
proof
set AT=AllTermsOf S,SS=AllSymbolsOf S,I=X-freeInterpreter(a),f=a-compound,
m=|.ar a.|, g=f|(m-tuples_on AT), AF=AtomicFormulasOf S, ch=chi(X,AF);
let T be m-element Element of AT*;
A1: dom f=(SS*\{{}})* by FUNCT_2:def 1;
A2: g.T = f.T & a-compound(T)=f.T by FOMODEL0:16, FUNCT_1:49, Def2;
thus a is not relational implies I.T=a-compound T
by A2, Def3; assume a is relational; then I=ch*g
by Def3 .= (ch*f)|(m-tuples_on AT) by RELAT_1:83;
then I.T=(ch*f).T by FUNCT_1:49, FOMODEL0:16 .= (ch.(f.T))
by FUNCT_1:13, A1; hence thesis by Def2;
end;
