theorem Th8: I-AtomicEval ((l,tt) AtomicSubst phi0) =
((l,I-TermEval.tt) ReassignIn I)-AtomicEval phi0
proof
set psi0=(l,tt) AtomicSubst phi0, u=I-TermEval.tt, J=(l,u) ReassignIn I,
F=S-firstChar, C=S-multiCat, FI=(S,{})-freeInterpreter, s1=F.phi0, s2=F.psi0,
n1=|.ar s1.|, n2=|.ar s2.|, TI=I-TermEval, TJ=J-TermEval, E=TheEqSymbOf S,
FJ=(l,tt) ReassignIn FI, d=U-deltaInterpreter;
not s1 in dom (l.-->({}.-->u)) by TARSKI:def 1; then
A1: s1=s2 & J.s1=I.s1 by FUNCT_4:11, Lm36;
A2: TI*(SubTerms psi0) = (TI*(FJ-TermEval*(SubTerms phi0))) by Lm36 .=
((TI*(FJ-TermEval))*(SubTerms phi0)) by RELAT_1:36 .=
(TJ*(SubTerms phi0)) by Lm37;
per cases;
suppose A3: s2<>E; then I-AtomicEval psi0 =
(J.s1).(TJ*(SubTerms phi0)) by FOMODEL2:14, A1, A2
.= J-AtomicEval phi0 by FOMODEL2:14, A3, A1; hence thesis;
end;
suppose A4: s2=E; then I-AtomicEval psi0= d.(TI*(SubTerms psi0))
by FOMODEL2:14 .= J-AtomicEval phi0 by FOMODEL2:14, A4, A1, A2;
hence thesis;
end;
end;
