let S be OrderSortedSign; :: thesis: for X being non-empty ManySortedSet of S holds union (rng (ParsedTerms X)) = TS (DTConOSA X)
let X be non-empty ManySortedSet of S; :: thesis: union (rng (ParsedTerms X)) = TS (DTConOSA X)
set D = DTConOSA X;
A1: dom (ParsedTerms X) = the carrier of S by PARTFUN1:def 2;
thus union (rng (ParsedTerms X)) c= TS (DTConOSA X) :: according to XBOOLE_0:def 10 :: thesis: TS (DTConOSA X) c= union (rng (ParsedTerms X))

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in union (rng (ParsedTerms X)) or x in TS (DTConOSA X) )
assume x in union (rng (ParsedTerms X)) ; :: thesis: x in TS (DTConOSA X)
then consider A being set such that
A2: x in A and
A3: A in rng (ParsedTerms X) by TARSKI:def 4;
consider s being object such that
A4: s in dom (ParsedTerms X) and
A5: (ParsedTerms X) . s = A by A3, FUNCT_1:def 3;
reconsider s = s as Element of S by A4;
A = ParsedTerms (X,s) by A5, Def8
.= { a where a is Element of TS (DTConOSA X) : ( ex s1 being Element of S ex x being object st
( s1 <= s & x in X . s1 & a = root-tree [x,s1] ) or ex o1 being OperSymbol of S st
( [o1, the carrier of S] = a . {} & the_result_sort_of o1 <= s ) ) } ;
then ex a being Element of TS (DTConOSA X) st
( a = x & ( ex s1 being Element of S ex x being object st
( s1 <= s & x in X . s1 & a = root-tree [x,s1] ) or ex o1 being OperSymbol of S st
( [o1, the carrier of S] = a . {} & the_result_sort_of o1 <= s ) ) ) by A2;
hence x in TS (DTConOSA X) ; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in TS (DTConOSA X) or x in union (rng (ParsedTerms X)) )
A6: the carrier of (DTConOSA X) = (Terminals (DTConOSA X)) \/ (NonTerminals (DTConOSA X)) by LANG1:1;
assume x in TS (DTConOSA X) ; :: thesis: x in union (rng (ParsedTerms X))
then reconsider t = x as Element of TS (DTConOSA X) ;
A7: rng t c= the carrier of (DTConOSA X) by RELAT_1:def 19;
A8: NonTerminals (DTConOSA X) = [: the carrier' of S,{ the carrier of S}:] by Th3;
A9: Terminals (DTConOSA X) = Union (coprod X) by Th3;
{} in dom t by TREES_1:22;
then A10: t . {} in rng t by FUNCT_1:def 3;

per cases ( t . {} in Terminals (DTConOSA X) or t . {} in NonTerminals (DTConOSA X) ) by A7, A10, A6, XBOOLE_0:def 3;

suppose A11: t . {} in Terminals (DTConOSA X) ; :: thesis: x in union (rng (ParsedTerms X))

then reconsider a = t . {} as Terminal of (DTConOSA X) ;
a in union (rng (coprod X)) by A9, A11, CARD_3:def 4;
then consider A being set such that
A12: a in A and
A13: A in rng (coprod X) by TARSKI:def 4;
consider s being object such that
A14: s in dom (coprod X) and
A15: (coprod X) . s = A by A13, FUNCT_1:def 3;
reconsider s = s as Element of S by A14;
A = coprod (s,X) by A15, MSAFREE:def 3;
then A16: ex b being set st
( b in X . s & a = [b,s] ) by A12, MSAFREE:def 2;
t = root-tree a by DTCONSTR:9;
then t in ParsedTerms (X,s) by A16;
then A17: t in (ParsedTerms X) . s by Def8;
(ParsedTerms X) . s in rng (ParsedTerms X) by A1, FUNCT_1:def 3;
hence x in union (rng (ParsedTerms X)) by A17, TARSKI:def 4; :: thesis: verum

end;

suppose t . {} in NonTerminals (DTConOSA X) ; :: thesis: x in union (rng (ParsedTerms X))

then reconsider a = t . {} as NonTerminal of (DTConOSA X) ;
consider o being Element of the carrier' of S, x2 being Element of { the carrier of S} such that
A18: a = [o,x2] by A8, DOMAIN_1:1;
set rs = the_result_sort_of o;
x2 = the carrier of S by TARSKI:def 1;
then t in ParsedTerms (X,(the_result_sort_of o)) by A18;
then A19: t in (ParsedTerms X) . (the_result_sort_of o) by Def8;
(ParsedTerms X) . (the_result_sort_of o) in rng (ParsedTerms X) by A1, FUNCT_1:def 3;
hence x in union (rng (ParsedTerms X)) by A19, TARSKI:def 4; :: thesis: verum

end;

end;