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 for I being (S,U)-interpreter-like Function,
phi being 0wff string of S st (S-firstChar.phi)=TheEqSymbOf S holds
(I-AtomicEval phi = 1 iff
(I-TermEval.((SubTerms phi).1) = I-TermEval.((SubTerms phi).2)))
proof
let I being (S,U)-interpreter-like Function, phi being 0wff string of S;
set E=TheEqSymbOf S, p=SubTerms phi, F=S-firstChar, s=F.phi, UV=I-TermEval,
V=I-AtomicEval phi,d=U-deltaInterpreter,U2=2-tuples_on U,TT=AllTermsOf S,D=
the set of all <*u,u*> where u is Element of U;
set n=|.ar s.|;
A1: U2=the set of all <*u1,u2*> where u1, u2 is Element of U
by FINSEQ_2:99;
A2: |.ar E.|-2=0; reconsider r=s as relational Element of S; set f=I===.r;
reconsider EE=E as relational Element of S; assume
A3: s=E; then V=d.(UV*p) & n=2 by Th14, A2; then
V=1 iff UV*p in d"{1} by FOMODEL0:25; then
A4: V=1 iff UV*p in D by Th13;
reconsider pp=p as 2-element FinSequence of TT by FINSEQ_1:def 11, A3, A2;
reconsider q=UV*pp as FinSequence of U;
reconsider qq=q as Element of U2 by FOMODEL0:16; qq in U2; then
consider u1, u2 being Element of U such that
A5: qq=<*u1,u2*> by A1; 1<=2 ; then 1<=1 & 1 <= len q
& 1<=2 & 2<=len q by CARD_1:def 7; then 1 in Seg (len q) & 2 in Seg
(len q); then 1 in dom q & 2 in dom q by FINSEQ_1:def 3; then
A7: q.1=UV.(p.1) & q.2=UV.(p.2) by FUNCT_1:12;
 q in D implies UV.(p.1)=UV.(p.2)
proof
assume q in D; then consider u being Element of U such that
A8: <*u,u*>=q; q.1=u & q.2=u by A8;
hence thesis by A7;
end;
hence thesis by A4, A7, A5;
end;
