set A = { a where a is Element of TS (DTConOSA X) : ( ex s1 being Element of S ex x being set st
( s1 <= s & x in X . s1 & a = root-tree [x,s1] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o <= s ) ) } ;
consider x being set such that
A1: x in X . s by XBOOLE_0:def 1;
set a = [x,s];
A2: [x,s] in coprod s,X by A1, MSAFREE:def 2;
A3: Terminals (DTConOSA X) = Union (coprod X) by Th3;
dom (coprod X) = the carrier of S by PARTFUN1:def 4;
then (coprod X) . s in rng (coprod X) by FUNCT_1:def 5;
then coprod s,X in rng (coprod X) by MSAFREE:def 3;
then [x,s] in union (rng (coprod X)) by A2, TARSKI:def 4;
then A4: [x,s] in Terminals (DTConOSA X) by A3, CARD_3:def 4;
then reconsider a = [x,s] as Symbol of (DTConOSA X) ;
reconsider b = root-tree a as Element of TS (DTConOSA X) by A4, DTCONSTR:def 4;
b in { a where a is Element of TS (DTConOSA X) : ( ex s1 being Element of S ex x being set st
( s1 <= s & x in X . s1 & a = root-tree [x,s1] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o <= s ) ) } by A1;
hence not ParsedTerms X,s is empty ; :: thesis: verum