theorem Th14:
for phi being 0wff string of S, I being (S,U)-interpreter-like Function
holds (S-firstChar.phi <> TheEqSymbOf S implies
I-AtomicEval phi=(I.(S-firstChar.phi)).((I-TermEval)*(SubTerms phi)))
& (S-firstChar.phi=TheEqSymbOf S implies
I-AtomicEval phi = (U-deltaInterpreter).((I-TermEval)*(SubTerms phi)))
proof
let phi be 0wff string of S, I be (S,U)-interpreter-like Function;
set TT=AllTermsOf S, E=TheEqSymbOf S, p=SubTerms phi, F=S-firstChar, r=F.phi,
n=|.ar r.|, AF=AtomicFormulasOf S, d=U-deltaInterpreter, p=SubTerms phi,
V=I-AtomicEval phi, f=(I===).r, UV=I-TermEval, G=I.r;
A1: |.ar E.|-2=0;
thus r<>E implies V=(I.(F.phi)).(UV*p)
proof
assume r <> E; then
not r in dom (E .--> d) by TARSKI:def 1;
hence V=G.(UV*p) by FUNCT_4:11;
end;
assume r=E; then
A2: r in {E} & n=2 by TARSKI:def 1, A1; then r in dom (E .--> d); then f = (E
.--> d).r by FUNCT_4:13 .= d by A2, FUNCOP_1:7;
hence V=d.(UV*p);
end;
