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;
