reserve k,m,n for Nat, kk,mm,nn for Element of NAT, A,B,X,Y,Z,x,y,z for set,
S, S1, S2 for Language, s for (Element of S), w,w1,w2 for (string of S),
U,U1,U2 for non empty set, f,g for Function, p,p1,p2 for FinSequence;
reserve u,u1,u2 for Element of U, t for termal string of S,
I for (S,U)-interpreter-like Function,
l, l1, l2 for literal (Element of S), m1, n1 for non zero Nat,
phi0 for 0wff string of S, psi,phi,phi1,phi2 for wff string of S;

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;
