set AL = (((FreeSort X) # ) * the Arity of S) . o;
set AX = ((FreeSort X) * the ResultSort of S) . o;
set D = DTConMSA X;
set O = the carrier' of S;
set rs = the_result_sort_of o;
set RS = the ResultSort of S;
defpred S₁[ set , set ] means for p being FinSequence of TS (DTConMSA X) st p = $1 holds
$2 = (Sym o,X) -tree p;
A1: for x being set st x in (((FreeSort X) # ) * the Arity of S) . o holds
ex y being set st
( y in ((FreeSort X) * the ResultSort of S) . o & S₁[x,y] )

proof

let x be set ; :: thesis: ( x in (((FreeSort X) # ) * the Arity of S) . o implies ex y being set st
( y in ((FreeSort X) * the ResultSort of S) . o & S₁[x,y] ) )
assume A2: x in (((FreeSort X) # ) * the Arity of S) . o ; :: thesis: ex y being set st
( y in ((FreeSort X) * the ResultSort of S) . o & S₁[x,y] )
then reconsider p = x as FinSequence of TS (DTConMSA X) by Th8;
Sym o,X ==> roots p by A2, Th10;
then reconsider a = (Sym o,X) -tree p as Element of TS (DTConMSA X) by DTCONSTR:def 4;
take y = (Sym o,X) -tree p; :: thesis: ( y in ((FreeSort X) * the ResultSort of S) . o & S₁[x,y] )
o in the carrier' of S ;
then o in dom ((FreeSort X) * the ResultSort of S) by PARTFUN1:def 4;
then A3: ((FreeSort X) * the ResultSort of S) . o = (FreeSort X) . (the ResultSort of S . o) by FUNCT_1:22
.= (FreeSort X) . (the_result_sort_of o) by MSUALG_1:def 7
.= FreeSort X,(the_result_sort_of o) by Def13 ;
a . {} = Sym o,X by TREES_4:def 4;
hence y in ((FreeSort X) * the ResultSort of S) . o by A3; :: thesis: S₁[x,y]
thus S₁[x,y] ; :: thesis: verum

end;

consider f being Function such that
A4: ( dom f = (((FreeSort X) # ) * the Arity of S) . o & rng f c= ((FreeSort X) * the ResultSort of S) . o & ( for x being set st x in (((FreeSort X) # ) * the Arity of S) . o holds
S₁[x,f . x] ) ) from WELLORD2:sch 1(A1);
reconsider g = f as Function of ((((FreeSort X) # ) * the Arity of S) . o),(rng f) by A4, FUNCT_2:3;
reconsider g = g as Function of ((((FreeSort X) # ) * the Arity of S) . o),(((FreeSort X) * the ResultSort of S) . o) by A4, FUNCT_2:4;
take g ; :: thesis: for p being FinSequence of TS (DTConMSA X) st Sym o,X ==> roots p holds
g . p = (Sym o,X) -tree p
let p be FinSequence of TS (DTConMSA X); :: thesis: ( Sym o,X ==> roots p implies g . p = (Sym o,X) -tree p )
assume Sym o,X ==> roots p ; :: thesis: g . p = (Sym o,X) -tree p
then p in (((FreeSort X) # ) * the Arity of S) . o by Th10;
hence g . p = (Sym o,X) -tree p by A4; :: thesis: verum