let S be locally_directed OrderSortedSign; :: thesis:
let X be V5() ManySortedSet of ; :: thesis:
set D = DTConOSA X;
set PTA = ParsedTermsOSA X;
set C = bool [:(TS (DTConOSA X)),the carrier of S:];
set SPTA = the Sorts of (ParsedTermsOSA X);
set F = PTClasses X;
defpred S₁[ set ] means for s being Element of S st ex y being Element of TS (DTConOSA X) st [y,s] in (PTClasses X) . $1 holds
[$1,s] in (PTClasses X) . $1;
A1: for s being Symbol of (DTConOSA X) st s in Terminals (DTConOSA X) holds
S₁[ root-tree s]

proof

let sy be Symbol of (DTConOSA X); :: thesis:
assume A2: sy in Terminals (DTConOSA X) ; :: thesis:
A3: (PTClasses X) . (root-tree sy) = @ sy by A2, Def22
.= { [(root-tree sy),s1] where s1 is Element of S : ex s2 being Element of S ex x being set st
( x in X . s2 & sy = [x,s2] & s2 <= s1 ) } ;
let s1 be Element of S; :: thesis:
assume ex y being Element of TS (DTConOSA X) st [y,s1] in (PTClasses X) . (root-tree sy) ; :: thesis:
then consider y being Element of TS (DTConOSA X) such that
A4: [y,s1] in (PTClasses X) . (root-tree sy) ;
consider s3 being Element of S such that
A5: [y,s1] = [(root-tree sy),s3] and
ex s2 being Element of S ex x being set st
( x in X . s2 & sy = [x,s2] & s2 <= s3 ) by A3, A4;
thus [(root-tree sy),s1] in (PTClasses X) . (root-tree sy) by A4, A5, ZFMISC_1:33; :: thesis:

end;

A6: for nt being Symbol of (DTConOSA X)
for ts being FinSequence of TS (DTConOSA X) st nt ==> roots ts & ( for t being DecoratedTree of st t in rng ts holds
S₁[t] ) holds
S₁[nt -tree ts]

proof

let nt be Symbol of (DTConOSA X); :: thesis:
let ts be FinSequence of TS (DTConOSA X); :: thesis:
assume that
A7: nt ==> roots ts and
for t being DecoratedTree of st t in rng ts holds
S₁[t] ; :: thesis:
consider o being OperSymbol of S such that
A8: ( nt = [o,the carrier of S] & ts in Args o,(ParsedTermsOSA X) & nt -tree ts = (Den o,(ParsedTermsOSA X)) . ts & ( for s1 being Element of S holds
( nt -tree ts in the Sorts of (ParsedTermsOSA X) . s1 iff the_result_sort_of o <= s1 ) ) ) by A7, Th12;
reconsider ts1 = ts as Element of Args o,(ParsedTermsOSA X) by A8;
set w = the_arity_of o;
reconsider x = (PTClasses X) * ts as FinSequence of bool [:(TS (DTConOSA X)),the carrier of S:] ;
A9: ( rng ts c= TS (DTConOSA X) & dom (PTClasses X) = TS (DTConOSA X) ) by FINSEQ_1:def 4, FUNCT_2:def 1;
then len x = len ts by FINSEQ_2:33;
then A10: ( dom x = dom ts & dom (the_arity_of o) = dom ts ) by A8, FINSEQ_3:31, MSUALG_3:6;
A11: for y being Nat st y in dom x holds
[(ts1 . y),((the_arity_of o) /. y)] in x . y

proof

let y be Nat; :: thesis:
assume A12: y in dom x ; :: thesis:
ts1 . y in rng ts1 by A10, A12, FUNCT_1:12;
then reconsider t1 = ts1 . y as Element of TS (DTConOSA X) by A9;
ts1 . y in the Sorts of (ParsedTermsOSA X) . ((the_arity_of o) /. y) by A10, A12, MSUALG_6:2;
then [t1,((the_arity_of o) /. y)] in (PTClasses X) . t1 by Th20;
hence [(ts1 . y),((the_arity_of o) /. y)] in x . y by A10, A12, FUNCT_1:23; :: thesis:

end;

A13: (PTClasses X) . (nt -tree ts) = @ nt,x by A7, Def22
.= { [((Den o2,(ParsedTermsOSA X)) . x2),s3] where o2 is OperSymbol of S, x2 is Element of Args o2,(ParsedTermsOSA X), s3 is Element of S : ( ex o1 being OperSymbol of S st
( nt = [o1,the carrier of S] & o1 ~= o2 & len (the_arity_of o1) = len (the_arity_of o2) & the_result_sort_of o1 <= s3 & the_result_sort_of o2 <= s3 ) & ex w3 being Element of the carrier of S * st
( dom w3 = dom x & ( for y being Nat st y in dom x holds
[(x2 . y),(w3 /. y)] in x . y ) ) ) } ;
let s1 be Element of S; :: thesis:
assume ex y being Element of TS (DTConOSA X) st [y,s1] in (PTClasses X) . (nt -tree ts) ; :: thesis:
then consider y being Element of TS (DTConOSA X) such that
A14: [y,s1] in (PTClasses X) . (nt -tree ts) ;
consider o2 being OperSymbol of S, x2 being Element of Args o2,(ParsedTermsOSA X), s3 being Element of S such that
A15: [y,s1] = [((Den o2,(ParsedTermsOSA X)) . x2),s3] and
A16: ( ex o1 being OperSymbol of S st
( nt = [o1,the carrier of S] & o1 ~= o2 & len (the_arity_of o1) = len (the_arity_of o2) & the_result_sort_of o1 <= s3 & the_result_sort_of o2 <= s3 ) & ex w3 being Element of the carrier of S * st
( dom w3 = dom x & ( for y being Nat st y in dom x holds
[(x2 . y),(w3 /. y)] in x . y ) ) ) by A13, A14;
A17: ( y = (Den o2,(ParsedTermsOSA X)) . x2 & s1 = s3 ) by A15, ZFMISC_1:33;
consider o1 being OperSymbol of S such that
A18: ( nt = [o1,the carrier of S] & o1 ~= o2 & len (the_arity_of o1) = len (the_arity_of o2) & the_result_sort_of o1 <= s3 & the_result_sort_of o2 <= s3 ) by A16;
( o ~= o & len (the_arity_of o) = len (the_arity_of o) & the_result_sort_of o <= s3 & the_result_sort_of o <= s3 ) by A8, A18, ZFMISC_1:33;
hence [(nt -tree ts),s1] in (PTClasses X) . (nt -tree ts) by A8, A10, A11, A13, A17; :: thesis:

end;

for t being DecoratedTree of st t in TS (DTConOSA X) holds
S₁[t] from DTCONSTR:sch 7(A1, A6);
hence for t being Element of TS (DTConOSA X)
for s being Element of S st ex y being Element of TS (DTConOSA X) st [y,s] in (PTClasses X) . t holds
[t,s] in (PTClasses X) . t ; :: thesis: