set A = { a where a is Element of TS (DTConMSA X) : ( ex x being set st
( x in X . s & a = root-tree [x,s] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o = s ) ) } ;
{ a where a is Element of TS (DTConMSA X) : ( ex x being set st
( x in X . s & a = root-tree [x,s] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o = s ) ) } c= TS (DTConMSA X)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { a where a is Element of TS (DTConMSA X) : ( ex x being set st
( x in X . s & a = root-tree [x,s] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o = s ) ) } or x in TS (DTConMSA X) )
assume x in { a where a is Element of TS (DTConMSA X) : ( ex x being set st
( x in X . s & a = root-tree [x,s] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o = s ) ) } ; :: thesis: x in TS (DTConMSA X)
then consider a being Element of TS (DTConMSA X) such that
A1: x = a and
( ex x being set st
( x in X . s & a = root-tree [x,s] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o = s ) ) ;
thus x in TS (DTConMSA X) by A1; :: thesis: verum

end;

hence { a where a is Element of TS (DTConMSA X) : ( ex x being set st
( x in X . s & a = root-tree [x,s] ) or ex o being OperSymbol of S st
( [o,the carrier of S] = a . {} & the_result_sort_of o = s ) ) } is Subset of (TS (DTConMSA X)) ; :: thesis: verum