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;
