let S be OrderSortedSign; :: thesis:
let X be V5() ManySortedSet of ; :: thesis:
A1: Union (coprod X) misses [:the carrier' of S,{the carrier of S}:] by MSAFREE:4;
set D = DTConOSA X;
set A = [:the carrier' of S,{the carrier of S}:] \/ (Union (coprod X));
A2: ( the carrier of (DTConOSA X) = (Terminals (DTConOSA X)) \/ (NonTerminals (DTConOSA X)) & Terminals (DTConOSA X) misses NonTerminals (DTConOSA X) ) by DTCONSTR:8, LANG1:1;
thus A3: NonTerminals (DTConOSA X) c= [:the carrier' of S,{the carrier of S}:] :: according to XBOOLE_0:def 10 :: thesis:

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in NonTerminals (DTConOSA X) ; :: thesis:
then x in { s where s is Symbol of (DTConOSA X) : ex n being FinSequence st s ==> n } by LANG1:def 3;
then consider s being Symbol of (DTConOSA X) such that
A4: ( s = x & ex n being FinSequence st s ==> n ) ;
consider n being FinSequence such that
A5: s ==> n by A4;
A6: [s,n] in the Rules of (DTConOSA X) by A5, LANG1:def 1;
reconsider s = s as Element of [:the carrier' of S,{the carrier of S}:] \/ (Union (coprod X)) ;
reconsider n = n as Element of ([:the carrier' of S,{the carrier of S}:] \/ (Union (coprod X))) * by A6, ZFMISC_1:106;
[s,n] in OSREL X by A5, LANG1:def 1;
hence x in [:the carrier' of S,{the carrier of S}:] by A4, Def4; :: thesis:

end;

thus A7: [:the carrier' of S,{the carrier of S}:] c= NonTerminals (DTConOSA X) :: thesis:

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume A8: x in [:the carrier' of S,{the carrier of S}:] ; :: thesis:
then consider o being Element of the carrier' of S, x2 being Element of {the carrier of S} such that
A9: x = [o,x2] by DOMAIN_1:9;
A10: the carrier of S = x2 by TARSKI:def 1;
then reconsider xa = [o,the carrier of S] as Element of the carrier of (DTConOSA X) by A8, A9, XBOOLE_0:def 3;
set O = the_arity_of o;
defpred S₁[ set , set ] means ex i being Element of S st
( i <= (the_arity_of o) /. $1 & $2 in coprod i,X );
A11: for a being set st a in Seg (len (the_arity_of o)) holds
ex b being set st S₁[a,b]

proof

let a be set ; :: thesis:
assume a in Seg (len (the_arity_of o)) ; :: thesis:
then A12: a in dom (the_arity_of o) by FINSEQ_1:def 3;
then A13: ( (the_arity_of o) . a in rng (the_arity_of o) & rng (the_arity_of o) c= the carrier of S ) by FINSEQ_1:def 4, FUNCT_1:def 5;
then not X . ((the_arity_of o) . a) is empty ;
then consider x being set such that
A14: x in X . ((the_arity_of o) . a) by XBOOLE_0:def 1;
take y = [x,((the_arity_of o) . a)]; :: thesis:
A15: y in coprod ((the_arity_of o) . a),X by A13, A14, MSAFREE:def 2;
take (the_arity_of o) /. a ; :: thesis:
thus ( (the_arity_of o) /. a <= (the_arity_of o) /. a & y in coprod ((the_arity_of o) /. a),X ) by A12, A15, PARTFUN1:def 8; :: thesis:

end;

consider b being Function such that
A16: ( dom b = Seg (len (the_arity_of o)) & ( for a being set st a in Seg (len (the_arity_of o)) holds
S₁[a,b . a] ) ) from CLASSES1:sch 1(A11);
reconsider b = b as FinSequence by A16, FINSEQ_1:def 2;
rng b c= [:the carrier' of S,{the carrier of S}:] \/ (Union (coprod X))

proof

let a be set ; :: according to TARSKI:def 3 :: thesis:
assume a in rng b ; :: thesis:
then consider c being set such that
A17: ( c in dom b & b . c = a ) by FUNCT_1:def 5;
consider i being Element of S such that
A18: ( i <= (the_arity_of o) /. c & a in coprod i,X ) by A16, A17;
dom (coprod X) = the carrier of S by PARTFUN1:def 4;
then (coprod X) . i in rng (coprod X) by FUNCT_1:def 5;
then coprod i,X in rng (coprod X) by MSAFREE:def 3;
then a in union (rng (coprod X)) by A18, 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;
A19: len b = len (the_arity_of o) by A16, FINSEQ_1:def 3;

now

let c be set ; :: thesis:
assume c in dom b ; :: thesis:
then consider i being Element of S such that
A20: ( i <= (the_arity_of o) /. c & b . c in coprod i,X ) by A16;
dom (coprod X) = the carrier of S by PARTFUN1:def 4;
then (coprod X) . i in rng (coprod X) by FUNCT_1:def 5;
then coprod i,X in rng (coprod X) by MSAFREE:def 3;
then b . c in union (rng (coprod X)) by A20, 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 A1, XBOOLE_0:3; :: thesis:
assume b . c in Union (coprod X) ; :: thesis:
thus ex i being Element of S st
( i <= (the_arity_of o) /. c & b . c in coprod i,X ) by A20; :: thesis:

end;

then [xa,b] in OSREL X by A19, Th2;
then xa ==> b by LANG1:def 1;
then xa in { t where t is Symbol of (DTConOSA X) : ex n being FinSequence st t ==> n } ;
hence x in NonTerminals (DTConOSA X) by A9, A10, LANG1:def 3; :: thesis:

end;

thus Terminals (DTConOSA X) c= Union (coprod X) :: according to XBOOLE_0:def 10 :: thesis:

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume A21: x in Terminals (DTConOSA X) ; :: thesis:
then A22: x in [:the carrier' of S,{the carrier of S}:] \/ (Union (coprod X)) by A2, XBOOLE_0:def 3;
not x in [:the carrier' of S,{the carrier of S}:] by A2, A7, A21, XBOOLE_0:3;
hence x in Union (coprod X) by A22, XBOOLE_0:def 3; :: thesis:

end;

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume A23: x in Union (coprod X) ; :: thesis:
then x in [:the carrier' of S,{the carrier of S}:] \/ (Union (coprod X)) by XBOOLE_0:def 3;
then ( x in Terminals (DTConOSA X) or x in NonTerminals (DTConOSA X) ) by A2, XBOOLE_0:def 3;
hence x in Terminals (DTConOSA X) by A1, A3, A23, XBOOLE_0:3; :: thesis: