set D = DTConMSA X;
set A = [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X));
A1: Terminals (DTConMSA X) = Union (coprod X) by Th6;
A2: NonTerminals (DTConMSA X) = [: the carrier' of S,{ the carrier of S}:] by Th6;
A3: Union (coprod X) misses [: the carrier' of S,{ the carrier of S}:] by Th4;
for nt being Symbol of (DTConMSA X) st nt in NonTerminals (DTConMSA X) holds
ex p being FinSequence of TS (DTConMSA X) st nt ==> roots p

let nt be Symbol of (DTConMSA X); :: thesis:
assume nt in NonTerminals (DTConMSA X) ; :: thesis:
then consider o being OperSymbol of S, x2 being Element of { the carrier of S} such that
A4: nt = [o,x2] by A2, DOMAIN_1:1;
set O = the_arity_of o;
defpred S₁[ object , object ] means X in coprod (((the_arity_of o) . S),X);
A5: for a being object st a in Seg (len (the_arity_of o)) holds
ex b being object st S₁[a,b]

let a be object ; :: thesis:
assume a in Seg (len (the_arity_of o)) ; :: thesis:
then a in dom (the_arity_of o) by FINSEQ_1:def 3;
then A6: (the_arity_of o) . a in rng (the_arity_of o) by FUNCT_1:def 3;
A7: rng (the_arity_of o) c= the carrier of S by FINSEQ_1:def 4;
then consider x being object such that
A8: x in X . ((the_arity_of o) . a) by A6, XBOOLE_0:def 1;
take [x,((the_arity_of o) . a)] ; :: thesis:
thus S₁[a,[x,((the_arity_of o) . a)]] by A6, A7, A8, Def2; :: thesis:

end;

consider b being Function such that
A9: ( dom b = Seg (len (the_arity_of o)) & ( for a being object st a in Seg (len (the_arity_of o)) holds
S₁[a,b . a] ) ) from CLASSES1:sch 1(A5);
reconsider b = b as FinSequence by A9, FINSEQ_1:def 2;
A10: rng b c= [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X))

let a be object ; :: according to TARSKI:def 3 :: thesis:
A11: rng (the_arity_of o) c= the carrier of S by FINSEQ_1:def 4;
assume a in rng b ; :: thesis:
then consider c being object such that
A12: c in dom b and
A13: b . c = a by FUNCT_1:def 3;
dom (the_arity_of o) = Seg (len (the_arity_of o)) by FINSEQ_1:def 3;
then A14: (the_arity_of o) . c in rng (the_arity_of o) by A9, A12, FUNCT_1:def 3;
dom (coprod X) = the carrier of S by PARTFUN1:def 2;
then (coprod X) . ((the_arity_of o) . c) in rng (coprod X) by A14, A11, FUNCT_1:def 3;
then A15: coprod (((the_arity_of o) . c),X) in rng (coprod X) by A14, A11, Def3;
a in coprod (((the_arity_of o) . c),X) by A9, A12, A13;
then a in union (rng (coprod X)) by A15, TARSKI:def 4;
then a in Union (coprod X) by CARD_3:def 4;
hence a in [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X)) by XBOOLE_0:def 3; :: thesis:

end;

then reconsider b = b as FinSequence of [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X)) by FINSEQ_1:def 4;
reconsider b = b as Element of ([: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X))) * by FINSEQ_1:def 11;
deffunc H₁( object ) -> set = root-tree (b . S);
consider f being Function such that
A16: ( dom f = dom b & ( for x being object st x in dom b holds
f . x = H₁(x) ) ) from FUNCT_1:sch 3();
reconsider f = f as FinSequence by A9, A16, FINSEQ_1:def 2;
rng f c= TS (DTConMSA X)

let x be object ; :: according to TARSKI:def 3 :: thesis:
A17: rng (the_arity_of o) c= the carrier of S by FINSEQ_1:def 4;
assume x in rng f ; :: thesis:
then consider y being object such that
A18: y in dom f and
A19: f . y = x by FUNCT_1:def 3;
dom (the_arity_of o) = Seg (len (the_arity_of o)) by FINSEQ_1:def 3;
then A20: (the_arity_of o) . y in rng (the_arity_of o) by A9, A16, A18, FUNCT_1:def 3;
dom (coprod X) = the carrier of S by PARTFUN1:def 2;
then (coprod X) . ((the_arity_of o) . y) in rng (coprod X) by A20, A17, FUNCT_1:def 3;
then A21: coprod (((the_arity_of o) . y),X) in rng (coprod X) by A20, A17, Def3;
b . y in rng b by A16, A18, FUNCT_1:def 3;
then reconsider a = b . y as Symbol of (DTConMSA X) by A10;
S₁[y,b . y] by A9, A16, A18;
then b . y in union (rng (coprod X)) by A21, TARSKI:def 4;
then A22: a in Terminals (DTConMSA X) by A1, CARD_3:def 4;
x = root-tree (b . y) by A16, A18, A19;
hence x in TS (DTConMSA X) by A22, DTCONSTR:def 1; :: thesis:

end;

then reconsider f = f as FinSequence of TS (DTConMSA X) by FINSEQ_1:def 4;
A23: for x being object st x in dom b holds
(roots f) . x = b . x

let x be object ; :: thesis:
assume A24: x in dom b ; :: thesis:
then reconsider i = x as Nat ;
( f . x = root-tree (b . x) & ex T being DecoratedTree st
( T = f . i & (roots f) . i = T . {} ) ) by A16, A24, TREES_3:def 18;
hence (roots f) . x = b . x by TREES_4:3; :: thesis:

end;

A25: now :: thesis:

let c be set ; :: thesis:
assume A26: c in dom b ; :: thesis:
dom (the_arity_of o) = Seg (len (the_arity_of o)) by FINSEQ_1:def 3;
then A27: (the_arity_of o) . c in rng (the_arity_of o) by A9, A26, FUNCT_1:def 3;
A28: rng (the_arity_of o) c= the carrier of S by FINSEQ_1:def 4;
dom (coprod X) = the carrier of S by PARTFUN1:def 2;
then (coprod X) . ((the_arity_of o) . c) in rng (coprod X) by A27, A28, FUNCT_1:def 3;
then A29: coprod (((the_arity_of o) . c),X) in rng (coprod X) by A27, A28, Def3;
S₁[c,b . c] by A9, A26;
then b . c in union (rng (coprod X)) by A29, TARSKI:def 4;
then b . c in Union (coprod X) by CARD_3:def 4;
hence ( b . c in [: the carrier' of S,{ the carrier of S}:] implies for o1 being OperSymbol of S st [o1, the carrier of S] = b . c holds
the_result_sort_of o1 = (the_arity_of o) . c ) by A3, XBOOLE_0:3; :: thesis:
assume b . c in Union (coprod X) ; :: thesis:
thus b . c in coprod (((the_arity_of o) . c),X) by A9, A26; :: thesis:

end;

( the carrier of S = x2 & len b = len (the_arity_of o) ) by A9, FINSEQ_1:def 3, TARSKI:def 1;
then [nt,b] in REL X by A4, A25, Th5;
then A30: nt ==> b by LANG1:def 1;
take f ; :: thesis:
dom (roots f) = dom f by TREES_3:def 18;
hence nt ==> roots f by A30, A16, A23, FUNCT_1:2; :: thesis:

end;

hence ( DTConMSA X is with_terminals & DTConMSA X is with_nonterminals & DTConMSA X is with_useful_nonterminals ) by A1, A2, DTCONSTR:def 3, DTCONSTR:def 4, DTCONSTR:def 5; :: thesis: