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;

theorem Th3:
((I,u)-TermEval.(m+1)).t =
(I.(S-firstChar.t)).(((I,u)-TermEval.m)*(SubTerms(t))) &
(t is 0-termal implies ((I,u)-TermEval.(m+1)).t = (I.(S-firstChar.t)).{})
proof
reconsider mm=m as Element of NAT by ORDINAL1:def 12;
((I,u)-TermEval.(mm+1)).t =
(I.(S-firstChar.t)).(((I,u)-TermEval.mm)*(SubTerms(t))) &
(t is 0-termal implies ((I,u)-TermEval.(mm+1)).t = (I.(S-firstChar.t)).{})
by Lm5;
hence thesis;
end;
