reserve k,m,n for Nat, kk,mm,nn for Element of NAT, X,Y,x,y,z for set;
reserve S,S1,S2 for Language, s,s1,s2 for Element of S;

theorem Th3: for w being (mm+1)-termal string of S ex T being Element of
((S-termsOfMaxDepth).mm)* st T is |.(ar(S-firstChar.w)).|-element
& w=<*S-firstChar.w*>^((S-multiCat).T)
proof
let w be (mm+1)-termal string of S; consider s being termal Element of S,
T being Element of ((S-termsOfMaxDepth).mm)* such that A1:
T is (|.ar s.|)-element & w=<*s*>^((S-multiCat).T) by Lm8;
reconsider ww=w as non empty FinSequence of (AllSymbolsOf S)
by FINSEQ_1:def 11;
s = w.1 by A1, FINSEQ_1:41 .= S-firstChar.w by FOMODEL0:6;
hence thesis by A1;
end;
