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 (TheEqSymbOf S1=TheEqSymbOf S2 & TheNorSymbOf S1=TheNorSymbOf S2)
implies for I1 being Element of U-InterpretersOf S1,
I2 being Element of U-InterpretersOf S2, phi1 being wff string of S1
st ((the adicity of S1)|(rng phi1/\OwnSymbolsOf S1) =
(the adicity of S2)|(rng phi1/\OwnSymbolsOf S1) &
I1|(rng phi1/\OwnSymbolsOf S1) = I2|(rng phi1/\OwnSymbolsOf S1) )
ex phi2 being wff string of S2 st phi1=phi2
proof
set O1=OwnSymbolsOf S1, O2=OwnSymbolsOf S2, a1=the adicity of S1,
a2=the adicity of S2, E1=TheEqSymbOf S1, E2=TheEqSymbOf S2, F1=S1-firstChar,
F2=S2-firstChar, AS1=AtomicFormulaSymbolsOf S1, AS2=AtomicFormulaSymbolsOf S2,
N1=TheNorSymbOf S1, N2=TheNorSymbOf S2, II1=U-InterpretersOf S1,
II2=U-InterpretersOf S2; assume
A1: E1=E2 & N1=N2;
defpred P[Nat] means for I1 being Element of II1, I2 being Element of II2,
phi1 being $1-wff string of S1 st a1|(rng phi1/\O1) = a2|(rng phi1/\O1) &
I1|(rng phi1/\O1) = I2|(rng phi1/\O1) ex phi2 being $1-wff string of S2 st
phi1=phi2;
A2: P[0]
proof
let I1 be Element of II1, I2 be Element of II2;
let phi1 be 0-wff string of S1; reconsider
phi11=phi1 as 0wff string of S1; set x1=rng phi1, x11=x1/\O1; assume
a1|x11 = a2|x11 & I1|x11=I2|x11; then
consider phi2 being 0wff string of S2 such that
A3: phi11=phi2 & I1-AtomicEval phi11=I2-AtomicEval phi2 by Lm48, A1;
thus thesis by A3;
end;
A4: for n st P[n] holds P[n+1]
proof
let n; set N=n+1; assume
A5: P[n]; let I1 be Element of II1, I2 be Element of II2;
let phi1 be N-wff string of S1; set x1=rng phi1, x11=x1/\O1; assume
A6: a1|x11 = a2|x11 & I1|x11=I2|x11;
per cases;
suppose phi1 is 0wff; then
reconsider phi11=phi1 as 0-wff string of S1;
consider phi2 being 0-wff string of S2 such that
A7: phi11=phi2 by A2, A6; phi2 is (0+0*N)-wff; then phi2 is (0+N)-wff; then
reconsider phi22=phi2 as N-wff string of S2; take phi22; thus thesis by A7;
end;
suppose not phi1 is 0wff; then reconsider phi11=phi1 as
non 0wff N-wff string of S1; reconsider h1=head phi11 as
n-wff string of S1; set t11=tail phi11, l11=F1.phi11;
A8: phi11=<*l11*>^h1^t11 by FOMODEL2:23; then
rng h1 c= rng (<*l11*>^h1) & rng (<*l11*>^h1) c= x1 by FINSEQ_1:30, 29;
then reconsider y1=rng h1 as non empty Subset of x1 by XBOOLE_1:1;
reconsider y11=y1/\O1 as Subset of x11 by XBOOLE_1:26;
A9: I1|(y11 null x11) = I1|x11|y11 by RELAT_1:71 .=
I2|(y11 null x11) by RELAT_1:71, A6;
a1|(y11 null x11) = a1|x11|y11 by RELAT_1:71 .=
a2|(y11 null x11) by RELAT_1:71, A6; then
consider h2 being n-wff string of S2 such that
A10: h1=h2 by A5, A9;
per cases;
suppose phi11 is exal; then reconsider phi11 as exal non 0wff N-wff string
of S1; reconsider l1=F1.phi11 as literal Element of S1;
phi1 null {} is (x1\/{})-valued; then {phi1.1} \ x1 = {}; then
phi1.1 in x1 by ZFMISC_1:60; then l1 in x1 & l1 in O1 &
dom a1=AS1 by FOMODEL0:6, FOMODEL1:def 19, FUNCT_2:def 1; then
A11: l1 in x11 & dom (a1|x11) = AS1/\x11 & l1 in AS1
by RELAT_1:61, XBOOLE_0:def 4, FOMODEL1:def 20; then
l1 in dom (a2|x11) & dom (a2|x11) = x11/\dom a2
by XBOOLE_0:def 4, RELAT_1:61, A6; then
l1 in dom a2 & dom a2=AS2 by FUNCT_2:def 1; then
reconsider l2=l1 as ofAtomicFormula Element of S2 by FOMODEL1:def 20;
l2 in O2 by A1, FOMODEL1:15; then
reconsider l2 as own Element of S2;
ar l1 =
a1|x11.l1 by A11, FUNCT_1:49 .=
ar l2 by A6, A11, FUNCT_1:49; then not l2 is low-compounding; then
reconsider l2 as literal Element of S2;
take phi2=<*l2*>^h2; phi11=<*l2*>^h1^(tail phi11) by FOMODEL2:23;
hence phi1=phi2 by A10;
reconsider l2 as literal Element of S2;
end;
suppose not phi11 is exal; then reconsider phi11 as non exal non 0wff
N-wff string of S1;
reconsider t1=tail phi11 as n-wff string of S1;
reconsider z1=rng t1 as non empty Subset of x1 by A8, FINSEQ_1:30;
reconsider z11=z1/\O1 as Subset of x11 by XBOOLE_1:26;
A12: I1|(z11 null x11) =
I2|x11|z11 by A6, RELAT_1:71 .= I2|(z11 null x11) by RELAT_1:71;
a1|(z11 null x11) = a1|x11|z11 by RELAT_1:71 .=
a2|(z11 null x11) by RELAT_1:71, A6; then
consider t2 being n-wff string of S2 such that
A13: t1=t2 by A5, A12; take phi2=<*N2*>^h2^t2;
F1.phi11 \+\ N1={};
hence phi1=phi2 by A1, A10, A13, A8, FOMODEL0:29;
end;
end;
end;
A14: for n holds P[n] from NAT_1:sch 2(A2, A4);
let I1 be Element of II1, I2 be Element of II2, phi1 be wff string of S1;
set d=Depth phi1; phi1 null 0 is (d+0)-wff; then
reconsider phi11=phi1 as d-wff string of S1;
set x1=rng phi1, x11=x1/\O1; assume a1|x11=a2|x11 & I1|x11=I2|x11; then
consider phi2 being d-wff string of S2 such that
A15: phi2=phi11 by A14; take phi2; thus thesis by A15;
end;
