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 Th9: (not l2 in rng psi) implies for I being Element of
U-InterpretersOf S holds (l1,u1) ReassignIn I-TruthEval psi=
(l2,u1) ReassignIn I-TruthEval (l1,l2)-SymbolSubstIn psi
proof
set II=U-InterpretersOf S,s=l1 SubstWith l2, SS=AllSymbolsOf S,SSS=SS\{l2},
TT=AllTermsOf S, T=S-termsOfMaxDepth, F=S-firstChar, N=TheNorSymbOf S;
reconsider SSSS=SSS as non empty Subset of SS;
defpred P[Nat] means for I being (Element of II), u, phi st
phi is $1-wff & phi is SSS-valued holds
((l1,u) ReassignIn I)-TruthEval phi =
((l2,u) ReassignIn I)-TruthEval ((l1,l2)-SymbolSubstIn phi);
A1: P[0]
proof
let I be Element of II; let u, phi;
set I1=(l1,u) ReassignIn I, I2=(l2,u) ReassignIn I;
assume phi is 0-wff; then reconsider phi0=phi as
0wff string of S; assume phi is SSS-valued; then
I1-TruthEval phi0 =
I2-TruthEval ((l1,l2)-SymbolSubstIn phi0) by Lm41; hence thesis;
end;
A2: for n st P[n] holds P[n+1]
proof
let n; assume
A3: P[n]; let I be Element of II; let u, phi;
set I1=(l1,u) ReassignIn I, I2=(l2,u) ReassignIn I; assume
A4: phi is (n+1)-wff & phi is SSS-valued; then reconsider
phii=phi as (n+1)-wff string of S;
reconsider x=phi as non empty SSSS-valued FinSequence by A4; {x.1} \ SSS={};
then phii.1 in SSS by ZFMISC_1:60; then F.phii in SSS by FOMODEL0:6; then
not F.phii in {l2} by XBOOLE_0:def 5; then
A5: F.phii <> l2 by TARSKI:def 1;
reconsider psi=(l1,l2)-SymbolSubstIn phii as (n+1)-wff string of S;
reconsider phi1=head phii as n-wff string of S;
reconsider psi1=(l1,l2)-SymbolSubstIn phi1 as n-wff string of S;
per cases;
suppose phi is exal & not phi is 0wff; then reconsider phii as
non 0wff exal (n+1)-wff string of S; set l=F.phii, phi2=tail phii;
A6: phii= <*l*>^phi1^phi2 by FOMODEL2:23 .= <*l*>^phi1; then
s.phii=s.<*l*>^(s.phi1) by FOMODEL0:36 .=
s.<*l*>^psi1 by FOMODEL0:def 22; then
A7: psi=s.<*l*>^psi1 by FOMODEL0:def 22;
x=<*l*>^phi1^{} by A6; then
A8: phi1 is SSSS-valued by FOMODEL0:44;
I1-TruthEval phii=1 iff I2-TruthEval psi=1
proof
per cases;
suppose A9: l=l1; then
A10: psi=<*l2*>^psi1 by A7, FOMODEL0:35;
hereby
assume I1-TruthEval phii=1; then consider u1 such that
A11: ((l,u1) ReassignIn I1)-TruthEval phi1=1 by A6, FOMODEL2:19;
1= ((l1,u1) ReassignIn I)-TruthEval phi1 by A11, A9, FOMODEL0:43 .=
((l2,u1) ReassignIn I)-TruthEval psi1 by A8, A3 .=
((l2,u1) ReassignIn I2)-TruthEval psi1 by FOMODEL0:43;
hence I2-TruthEval psi=1 by A10, FOMODEL2:19;
end;
assume I2-TruthEval psi=1; then consider u2 such that
A12: ((l2,u2) ReassignIn I2)-TruthEval psi1=1 by A10, FOMODEL2:19;
1 = ((l2,u2) ReassignIn I)-TruthEval psi1 by A12, FOMODEL0:43 .=
((l1,u2) ReassignIn I)-TruthEval phi1 by A8, A3 .=
((l,u2) ReassignIn I1)-TruthEval phi1 by A9, FOMODEL0:43;
hence I1-TruthEval phii=1 by A6, FOMODEL2:19;
end;
suppose A13: l<>l1; then
A14: psi=<*l*>^psi1 by A7, FOMODEL0:35;
hereby
assume I1-TruthEval phii=1; then consider u1 such that
A15: ((l,u1) ReassignIn I1)-TruthEval phi1=1 by A6, FOMODEL2:19;
1 = ((l1,u) ReassignIn (l,u1) ReassignIn I)-TruthEval phi1
by A15, A13, FOMODEL0:43
.= ((l2,u) ReassignIn (l,u1) ReassignIn I)-TruthEval psi1 by A3, A8
.= ((l,u1) ReassignIn (l2,u) ReassignIn I)-TruthEval psi1
by A5, FOMODEL0:43;
hence I2-TruthEval psi=1 by A14, FOMODEL2:19;
end;
assume I2-TruthEval psi=1; then consider u2 such that
A16: ((l,u2) ReassignIn I2)-TruthEval psi1=1 by A14, FOMODEL2:19;
1 = ((l2,u) ReassignIn (l,u2) ReassignIn I)-TruthEval psi1
by A5, A16, FOMODEL0:43 .=
((l1,u) ReassignIn (l,u2) ReassignIn I)-TruthEval phi1 by A3, A8
.= ((l,u2) ReassignIn (l1,u) ReassignIn I)-TruthEval phi1
by A13, FOMODEL0:43; hence I1-TruthEval phii=1 by A6, FOMODEL2:19;
end;
end;
then I1-TruthEval phii=1 iff not I2-TruthEval psi=0 by FOMODEL0:39;
hence thesis by FOMODEL0:39;
end;
suppose
not phi is exal & not phi is 0wff; then reconsider phii
as (n+1)-wff non exal non 0wff string of S;
reconsider phi2=tail phii as n-wff string of S;
reconsider psi2=(l1,l2)-SymbolSubstIn phi2
as n-wff string of S; F.phii \+\ N={}; then F.phii=N by FOMODEL0:29; then
A17: phii=<*N*>^phi1^phi2 by FOMODEL2:23;then phi1 is SSS-valued &
phi2 is SSS-valued & (I1-TruthEval phii=1 iff
(I1-TruthEval phi1=0 & I1-TruthEval phi2=0))
by A4, FOMODEL0:44, FOMODEL2:19; then
A18:(I1-TruthEval phii=1 iff (I2-TruthEval psi1=0 & I2-TruthEval psi2=0))
by A3;
A19: s.phii=psi & s.phi1=psi1 & s.phi2=psi2
by FOMODEL0:def 22; then psi=s.(<*N*>^phi1) ^ (s.phi2)
by A17, FOMODEL0:36 .= s.<*N*>^(s.phi1)^(s.phi2) by FOMODEL0:36
.= <*N*>^psi1^psi2 by FOMODEL0:35, A19; then
I2-TruthEval psi=1 iff not I1-TruthEval phi=0
by FOMODEL0:39, FOMODEL2:19, A18; hence thesis by FOMODEL0:39;
end;
suppose phi is 0-wff; hence thesis by A1, A4;
end;
end;
A20: for m holds P[m] from NAT_1:sch 2(A1, A2);
set m=Depth psi; assume not l2 in rng psi; then
{l2} misses rng psi & rng psi c= SS by RELAT_1:def 19, ZFMISC_1:50; then
A21: psi is m-wff & psi is SSS-valued
by XBOOLE_1:86, FOMODEL2:def 31, RELAT_1:def 19;
let I be Element of II; thus thesis by A20, A21;
end;
