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;

theorem Th17:
Depth(<*TheNorSymbOf S*>^phi1^phi2)=1+max(Depth phi1, Depth phi2) &
Depth (<*l*>^phi1) = Depth phi1 + 1
proof
set N=TheNorSymbOf S, m1=Depth phi1, m2=Depth phi2,
e=<*l*>^phi1, n=<*N*>^phi1^phi2; thus Depth n=1+max(m1,m2) by Lm31;
now
let m; assume C0: e is m-wff; assume m<(m1+1); then m<=m1 by NAT_1:13;
then m-m <= m1-m by XREAL_1:13; then
reconsider k=m1-m as Nat; e is (m+0*k)-wff by C0; then e is (m+k)-wff;
hence contradiction;
end; hence
Depth e=m1+1 by Def30;
end;
