reserve k,m,n for Nat, kk,mm,nn for Element of NAT, A,B,X,Y,Z,x,y,z for set,
S, S1, S2 for Language, s for (Element of S), w,w1,w2 for (string of S),
U,U1,U2 for non empty set, f,g for Function, p,p1,p2 for FinSequence;
reserve u,u1,u2 for Element of U, t for termal string of S,
I for (S,U)-interpreter-like Function,
l, l1, l2 for literal (Element of S), m1, n1 for non zero Nat,
phi0 for 0wff string of S, psi,phi,phi1,phi2 for wff string of S;
reserve I for Element of U-InterpretersOf S;
reserve I for (S,U)-interpreter-like Function;

theorem Th20:
(I,u)-TermEval.(m+1)|(S-termsOfMaxDepth.m) =I-TermEval|(S-termsOfMaxDepth.m)
proof
reconsider mm=m, MM=m+1 as Element of NAT by ORDINAL1:def 12;
set T=S-termsOfMaxDepth, TI=I-TermEval, TII=(I,u)-TermEval, TT=AllTermsOf S;
reconsider IM=TII.MM as Function of TT,U;
reconsider Tm=T.mm, TM=T.MM as Subset of TT by FOMODEL1:2;
set LH=IM|Tm, RH=TI|Tm;
A1: dom LH = Tm & dom RH = Tm by PARTFUN1:def 2;
now
let x be object; assume
A2: x in dom LH; then x in Tm null TT; then
reconsider tt=x as Element of TT; reconsider ttt=x as Element of Tm by A2;
LH.ttt\+\IM.ttt={} & RH.ttt\+\TI.ttt={}; then
A3: LH.x=IM.tt & RH.x=TI.x by FOMODEL0:29; then LH.x = I-TermEval tt
by A2,Def8 .= RH.x by A3, Def9;
hence LH.x=RH.x;
end;
hence thesis by A1, FUNCT_1:2;
end;
