let S be non empty non void ManySortedSign ; :: thesis:
let X be ManySortedSet of the carrier of S; :: thesis:
set D = DTConMSA X;
set A = [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X));
A1: the carrier of (DTConMSA X) = (Terminals (DTConMSA X)) \/ (NonTerminals (DTConMSA X)) by LANG1:1;
thus A2: NonTerminals (DTConMSA X) c= [: the carrier' of S,{ the carrier of S}:] :: thesis:

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in NonTerminals (DTConMSA X) ; :: thesis:
then x in { s where s is Symbol of (DTConMSA X) : ex n being FinSequence st s ==> n } by LANG1:def 3;
then consider s being Symbol of (DTConMSA X) such that
A3: s = x and
A4: ex n being FinSequence st s ==> n ;
consider n being FinSequence such that
A5: s ==> n by A4;
[s,n] in the Rules of (DTConMSA X) by A5, LANG1:def 1;
then reconsider n = n as Element of ([: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X))) * by ZFMISC_1:87;
reconsider s = s as Element of [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X)) ;
[s,n] in REL X by A5, LANG1:def 1;
hence x in [: the carrier' of S,{ the carrier of S}:] by A3, Def7; :: thesis:

end;

A6: Union (coprod X) misses [: the carrier' of S,{ the carrier of S}:] by Th4;
thus A7: Union (coprod X) c= Terminals (DTConMSA X) :: thesis:

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A8: 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 (DTConMSA X) or x in NonTerminals (DTConMSA X) ) by A1, XBOOLE_0:def 3;
hence x in Terminals (DTConMSA X) by A6, A2, A8, XBOOLE_0:3; :: thesis:

end;

assume A9: X is non-empty ; :: thesis:
thus NonTerminals (DTConMSA X) c= [: the carrier' of S,{ the carrier of S}:] by A2; :: according to XBOOLE_0:def 10 :: thesis:
thus A10: [: the carrier' of S,{ the carrier of S}:] c= NonTerminals (DTConMSA X) :: thesis:

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A11: x in [: the carrier' of S,{ the carrier of S}:] ; :: thesis:
then consider o being OperSymbol of S, x2 being Element of { the carrier of S} such that
A12: x = [o,x2] by DOMAIN_1:1;
set O = the_arity_of o;
defpred S₁[ object , object ] means $2 in coprod (((the_arity_of o) . $1),X);
A13: 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 A14: (the_arity_of o) . a in rng (the_arity_of o) by FUNCT_1:def 3;
A15: rng (the_arity_of o) c= the carrier of S by FINSEQ_1:def 4;
then consider x being object such that
A16: x in X . ((the_arity_of o) . a) by A9, A14, XBOOLE_0:def 1;
take [x,((the_arity_of o) . a)] ; :: thesis:
thus S₁[a,[x,((the_arity_of o) . a)]] by A14, A15, A16, Def2; :: thesis:

end;

consider b being Function such that
A17: ( 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(A13);
reconsider b = b as FinSequence by A17, FINSEQ_1:def 2;
rng b c= [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X))

let a be object ; :: according to TARSKI:def 3 :: thesis:
A18: 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
A19: c in dom b and
A20: 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 A21: (the_arity_of o) . c in rng (the_arity_of o) by A17, A19, 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 A21, A18, FUNCT_1:def 3;
then A22: coprod (((the_arity_of o) . c),X) in rng (coprod X) by A21, A18, Def3;
a in coprod (((the_arity_of o) . c),X) by A17, A19, A20;
then a in union (rng (coprod X)) by A22, 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;

A23: now :: thesis:

let c be set ; :: thesis:
assume A24: c in dom b ; :: thesis:
dom (the_arity_of o) = Seg (len (the_arity_of o)) by FINSEQ_1:def 3;
then A25: (the_arity_of o) . c in rng (the_arity_of o) by A17, A24, FUNCT_1:def 3;
A26: 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 A25, A26, FUNCT_1:def 3;
then A27: coprod (((the_arity_of o) . c),X) in rng (coprod X) by A25, A26, Def3;
S₁[c,b . c] by A17, A24;
then b . c in union (rng (coprod X)) by A27, 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 A6, XBOOLE_0:3; :: thesis:
assume b . c in Union (coprod X) ; :: thesis:
thus b . c in coprod (((the_arity_of o) . c),X) by A17, A24; :: thesis:

end;

A28: the carrier of S = x2 by TARSKI:def 1;
then reconsider xa = [o, the carrier of S] as Element of the carrier of (DTConMSA X) by A11, A12, XBOOLE_0:def 3;
len b = len (the_arity_of o) by A17, FINSEQ_1:def 3;
then [xa,b] in REL X by A23, Th5;
then xa ==> b by LANG1:def 1;
then xa in { t where t is Symbol of (DTConMSA X) : ex n being FinSequence st t ==> n } ;
hence x in NonTerminals (DTConMSA X) by A12, A28, LANG1:def 3; :: thesis:

end;

A29: Terminals (DTConMSA X) misses NonTerminals (DTConMSA X) by DTCONSTR:8;
thus Terminals (DTConMSA X) c= Union (coprod X) :: according to XBOOLE_0:def 10 :: thesis:

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in Terminals (DTConMSA X) ; :: thesis:
then ( x in [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X)) & not x in [: the carrier' of S,{ the carrier of S}:] ) by A1, A29, A10, XBOOLE_0:3, XBOOLE_0:def 3;
hence x in Union (coprod X) by XBOOLE_0:def 3; :: thesis:

end;

thus Union (coprod X) c= Terminals (DTConMSA X) by A7; :: thesis: