reserve A,B,C,X,Y,Z,x,x1,x2,y,z for set, U,U1,U2,U3 for non empty set,
u,u1,u2 for (Element of U), P,Q,R for Relation, f,g for Function,
k,m,n for Nat, m1, n1 for non zero Nat, kk,mm,nn for (Element of NAT),
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;
reserve S, S1, S2 for Language, s,s1,s2 for Element of S,
l,l1,l2 for literal Element of S, a for ofAtomicFormula Element of S,
r for relational Element of S, w,w1,w2 for string of S,
t,t1,t2 for termal string of S;
reserve phi0 for 0wff string of S,
psi, psi1, psi2, phi,phi1,phi2 for wff string of S,
I for (S,U)-interpreter-like Function;
reserve tt,tt0,tt1,tt2 for Element of AllTermsOf S;

theorem for I being Element of U-InterpretersOf S st l is X-absent &
X is I-satisfied holds X is (l,u) ReassignIn I-satisfied
proof
set II=U-InterpretersOf S; let I be Element of II; set O=OwnSymbolsOf S,
I2=(l,u) ReassignIn I, f2=l.-->({}.-->u); assume
A1: l is X-absent & X is I-satisfied;
now
let phi; reconsider no=rng phi/\O as Subset of rng phi; assume
A2: phi in X; then reconsider Phi={phi} as Subset of X by ZFMISC_1:31;
A3: I-TruthEval phi=1 by A1, A2;
l is (X/\Phi)-absent by A1; then not l in no by FOMODEL2:28; then
{l} misses no by ZFMISC_1:50; then dom f2 misses no; then
I|no +* (f2|no) = I|no null {} by RELAT_1:66; then
I2|no=I|no by FUNCT_4:71; hence I2-TruthEval phi=1 by A3, Th13;
end; hence thesis;
end;
