theorem Th16: for X being D1-expanded set st  ::# Henkin's theorem
R#1(S) in D1 & R#4(S) in D1 & R#6(S) in D1 & R#7(S) in D1 &
R#8(S) in D1 & X is S-mincover & X is S-witnessed
holds (D1 Henkin X)-TruthEval psi = 1 iff psi in X
proof
let X be D1-expanded set; set TT=AllTermsOf S, E=TheEqSymbOf S,
F=S-firstChar, N=TheNorSymbOf S, R=(X,D1)-termEq, U=Class R, L=LettersOf S,
AF=AtomicFormulasOf S, d=U-deltaInterpreter, i=(S,X)-freeInterpreter,
II=U-InterpretersOf S, D=D1, ii=TT-InterpretersOf S, G0=R#0(S), G1=R#1(S),
G2=R#2(S), G4=R#4(S), G6=R#6(S), G7=R#7(S), G8=R#8(S), E0={G0}, E1={G1},
E2={G2}, E4={G4}, E6={G6}, E7={G7}, E8={G8}; reconsider
E0, E1, E2, E4, E6, E7, E8 as RuleSet of S; assume
G1 in D & G4 in D & G6 in D & G7 in D & G8 in D; then
G0 in D & G1 in D & G2 in D & G4 in D & G6 in D & G7 in D & G8 in D by
Def62; then
reconsider F0=E0, F1=E1, F2=E2, F4=E4, F6=E6, F7=E7, F8=E8 as Subset of D
by ZFMISC_1:31;
A1: (F0\/(F0\/F1\/F8)) c= D & F0\/F6 c= D & F0 c= D &
F0\/(F0\/F1\/F8\/F7) c= D;
reconsider I=D1 Henkin X as Element of II;
set UV=I-TermEval, uv=i-TermEval, O=OwnSymbolsOf S, FF=AllFormulasOf S,
C=S-multiCat, SS=AllSymbolsOf S; assume
A2: X is S-mincover & X is S-witnessed;
defpred P[Nat] means for phi st phi is $1-wff holds
(I-TruthEval phi=1 iff phi in X);
A3: P[0]
proof
let phi; assume phi is 0-wff; then reconsider phi0=phi as 0wff string of S;
I-AtomicEval phi0=1 iff phi0 in X by Lm50; hence thesis;
end;
A4: for n st P[n] holds P[n+1]
proof
let n; set Vn=(I,n)-TruthEval; assume
A5: P[n]; let phi; set s=F.phi, V=I-TruthEval phi; assume
A6: phi is (n+1)-wff;
per cases;
suppose phi is non 0wff & phi is exal; then
reconsider phii=phi as non 0wff exal (n+1)-wff string of S by A6;
reconsider phi1=head phii as n-wff string of S;
reconsider l=F.phii as literal Element of S;
A7: phii=<*l*>^phi1^(tail phii) by FOMODEL2:23 .= <*l*>^phi1;
hereby
assume V=1;
then consider u being Element of U such that
A8: ((l,u) ReassignIn I)-TruthEval phi1=1 by A7, FOMODEL2:19;
consider x being object such that
A9: x in TT & u=Class (R,x) by EQREL_1:def 3;
reconsider tt=x as Element of TT by A9;
reconsider psi1=(l,tt) SubstIn phi1 as n-wff string of S; id TT.tt=tt & ((R
-class)*(i-TermEval)).tt \+\
(R-class).(i-TermEval.tt)={}; then
A10: i-TermEval.tt=tt & ((R-class)*(i-TermEval)).tt=(R-class).(i-TermEval.tt)
by FOMODEL0:29, FOMODEL3:4; I-TermEval.tt = ((R-class)*(i-TermEval)).tt
by FOMODEL3:3 .= u by A10, FOMODEL3:def 13, A9; then
1 = I-TruthEval psi1 by A8, FOMODEL3:10; then
A11: psi1 in X by A5;
[{(l,tt) SubstIn phi1},<*l*>^phi1] is
(1,{},{R#4(S)})-derivable; then <*l*>^phi1 is (X,E4)-provable &
F4 c= D & E4 is isotone by A11, ZFMISC_1:31; then
phii is (X,D)-provable by A7, Lm19; hence phi in X by Def18;
end;
assume phi in X; then consider l2 such that
A12: (l,l2)-SymbolSubstIn phi1 in X & not l2 in rng phi1 by A2, A7;
reconsider psi1=(l,l2)-SymbolSubstIn phi1 as n-wff string of S;
consider u being Element of U such that
A13: u=I.l2.{} & (l2,u) ReassignIn I=I by FOMODEL2:26;
reconsider I2=(l2,u) ReassignIn I, I1=(l,u) ReassignIn I as Element of II;
I2-TruthEval psi1=1 by A12, A5, A13; then
I1-TruthEval phi1=1 by A12, FOMODEL3:9; hence thesis by A7, FOMODEL2:19;
end;
suppose phi is non 0wff & phi is non exal; then
reconsider phii=phi as non 0wff non exal (n+1)-wff string of S by A6;
set phi1=head phii, phi2=tail phii; F.phii\+\N={}; then
s = N by FOMODEL0:29; then
A14: phi=<*N*>^phi1^phi2 by FOMODEL2:23;
V=1 iff (I-TruthEval phi1=0 & I-TruthEval phi2=0) by A14, FOMODEL2:19;
then V=1 iff ((not I-TruthEval phi1=1) & (not I-TruthEval phi2=1)) by
FOMODEL0:39; then
A15: V=1 iff ((not phi1 in X) & (not phi2 in X)) by A5;
A16: now
assume xnot phi1 in X & xnot phi2 in X; then xnot phi1 is
(X,{R#0(S)})-provable & xnot phi2 is (X,{R#0(S)})-provable by Th6; then
xnot phi1 is (X,D1)-provable & xnot phi2 is (X,D1)-provable by A1, Lm19;
then xnot phi1 in X & xnot phi2 in X by Def18; then
reconsider Y={xnot phi1, xnot phi2} as Subset of X by ZFMISC_1:32;
phi is (X null Y,D1)-provable by Lm19, A1, A14;
hence phi in X by Def18;
end;
now
reconsider H={phi} as S-premises-like set; assume phi in X; then
E7: H c= X by ZFMISC_1:31;
A17: [{phi},phi] is (1, {}, E0)-derivable;
A18: [H null phi2,xnot phi1] is (2,{[{phi},phi]},E0\/E1\/E8)-derivable by A14;
A19: [H null phi1, xnot phi2] is (3,{[H,phi]},E0\/E1\/E8\/E7)-derivable by A14;
[H,xnot phi1] is (1+2,{},E0\/(E0\/E1\/E8))-derivable by A18, Lm22; then
[H,xnot phi1] is (3,{},D)-derivable by A1, Th2; then
xnot phi1 is (X,D)-provable by E7;
hence xnot phi1 in X by Def18;
[H,xnot phi2] is (1+3, {}, E0\/(E0\/E1\/E8\/E7))-derivable
by A17, A19, Lm22; then
[H,xnot phi2] is (4, {}, D)-derivable by A1, Th2; then
xnot phi2 is (X,D)-provable by E7;
hence xnot phi2 in X by Def18;
end;
hence thesis by A15, A2, A16;
end;
suppose phi is 0wff; hence thesis by A3; end;
end;
A21: for n holds P[n] from NAT_1:sch 2(A3, A4); psi is (Depth psi)-wff
by FOMODEL2:def 31;
hence thesis by A21;
end;
