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 Th13: for I1, I2 being Element of U-InterpretersOf S st
I1|(rng phi/\OwnSymbolsOf S)=I2|(rng phi/\OwnSymbolsOf S) holds
I1-TruthEval phi=I2-TruthEval phi
proof
set O=OwnSymbolsOf S, II=U-InterpretersOf S, a=the adicity of S,
E=TheEqSymbOf S, F=S-firstChar, C=S-multiCat;
defpred P[Nat] means for I1,I2 being Element of II,
phi being $1-wff string of S st I1|(rng phi/\O)=I2|(rng phi/\O) holds
I1-TruthEval phi=I2-TruthEval phi;
A1: P[0]
proof
let I1, I2 be Element of II; let phi be 0-wff string of S;
reconsider phi1=phi as 0wff string of S;
assume I1|(rng phi/\O)=I2|(rng phi/\O); then
I1|(rng phi1/\O)=I2|(rng phi1/\O) & a|(rng phi1/\O)=a|(rng phi1/\O) & E=E;
then consider phi2 being 0wff string of S such that
A2: phi2=phi1 & I2-AtomicEval phi2=I1-AtomicEval phi1 by Lm48;
thus thesis by A2;
end;
A3: for n st P[n] holds P[n+1]
proof
let n; assume
A4: P[n]; let I1, I2 be Element of II;
let phi be (n+1)-wff string of S; assume
A5: I1|(rng phi/\O) = I2|(rng phi/\O);
per cases;
suppose not phi is 0wff & not phi is exal; then reconsider phii=phi as
non 0wff non exal (n+1)-wff string of S; set X=rng phii/\O, s=F.phii;
reconsider h=head phii, t=tail phii as n-wff string of S;
phii=<*s*>^h^t by FOMODEL2:23 .= <*s*>^(h^t) by FINSEQ_1:32; then
rng (h^t) c= rng phii & rng t c= rng (h^t) & rng h c= rng (h^t)
by FINSEQ_1:29, 30; then rng h c= rng phii & rng t c= rng phii
by XBOOLE_1:1; then reconsider rh=(rng h/\O), rt=(rng t/\O) as Subset of X
by XBOOLE_1:26; set v1=I1-TruthEval phii, v2=I2-TruthEval phii,
h1=I1-TruthEval h, h2=I2-TruthEval h, t1=I1-TruthEval t, t2=I2-TruthEval t;
A6: I1|rh=I1|(rh null X) .= I1|X|rh by RELAT_1:71 .=
I2|(rh null X) by A5, RELAT_1:71;
I1|rt = I1|(rt null X) .= I1|X|rt by RELAT_1:71 .=
I2|(rt null X) by A5, RELAT_1:71; then
A7: t1 = t2 by A4;
v1 \+\ (h1 'nor' t1) = {} & v2 \+\ (h2 'nor' t2)={}; then v1=h1 'nor' t1
& v2=h2 'nor' t2 by FOMODEL0:29;
hence thesis by A4, A6, A7;
end;
suppose phi is exal & not phi is 0wff; then reconsider phii=phi
as exal wff string of S; set l=F.phii; reconsider
h=head phii as n-wff string of S;
A8: phii=<*l*>^h^(tail phii) by FOMODEL2:23 .= <*l*>^h; then
reconsider rh=rng h as Subset of rng phii by FINSEQ_1:30;
now
hereby
assume I1-TruthEval phii=1; then consider u such that
A9: (l,u) ReassignIn I1-TruthEval h=1 by A8, FOMODEL2:19;
set f=l.-->({}.-->u); reconsider
I1u=(l,u) ReassignIn I1, I2u=(l,u) ReassignIn I2 as Element of II;
I1u|(rng h/\O) = I1|(rh null (rng phii)/\O) +* f|(rh/\O) by FUNCT_4:71 .=
I1|(rh/\(rng phii/\O)) +* f|(rh/\O) by XBOOLE_1:16 .=
I1|(rng phii/\O)|rh +* f|(rh/\O) by RELAT_1:71 .=
I2|(rng phii/\O/\rh) +* f|(rh/\O) by RELAT_1:71, A5 .=
I2|(rng phii/\rh/\O) +* f|(rh/\O) by XBOOLE_1:16 .=
I2u|(rng h/\O) by FUNCT_4:71; then I2u-TruthEval h=1 by A9, A4;
hence I2-TruthEval phii=1 by A8, FOMODEL2:19;
end;
assume I2-TruthEval phii=1; then consider u such that
A10: (l,u) ReassignIn I2-TruthEval h=1 by A8, FOMODEL2:19;
set f=l.-->({}.-->u); reconsider
I1u=(l,u) ReassignIn I1, I2u=(l,u) ReassignIn I2 as Element of II;
I1u|(rng h/\O) = I1|(rh null (rng phii)/\O) +* f|(rh/\O) by FUNCT_4:71 .=
I1|(rh/\(rng phii/\O)) +* f|(rh/\O) by XBOOLE_1:16 .=
I1|(rng phii/\O)|rh +* f|(rh/\O) by RELAT_1:71 .=
I2|(rng phii/\O/\rh) +* f|(rh/\O) by RELAT_1:71, A5 .=
I2|(rng phii/\rh/\O) +* f|(rh/\O) by XBOOLE_1:16 .=
I2u|(rng h/\O) by FUNCT_4:71; then
I1u-TruthEval h=1 by A10, A4; hence I1-TruthEval phii=1 by A8, FOMODEL2:19;
end; then
I1-TruthEval phii=1 iff not I2-TruthEval phii=0 by FOMODEL0:39;
hence thesis by FOMODEL0:39;
end;
suppose phi is 0wff; hence thesis by A1,A5;
end;
end;
A11: for n holds P[n] from NAT_1:sch 2(A1, A3); let I1, I2 be Element of II;
set d=Depth phi; phi null 0 is (d+0)-wff; then
reconsider phii=phi as d-wff string of S;
assume I1|(rng phi/\O)=I2|(rng phi/\O); then
I1|(rng phii/\O)=I2|(rng phii/\O); hence thesis by A11;
end;
