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;
