defpred S1[ Element of HP-WFF ] means HP-Subformulae . $1 is DecoratedTree of HP-WFF ;
A1: for n being Element of NAT holds S1[ prop n]
proof
let n be Element of NAT ; :: thesis: S1[ prop n]
HP-Subformulae . (prop n) = root-tree (prop n) by Def9;
hence S1[ prop n] ; :: thesis: verum
end;
A2: for r, s being Element of HP-WFF st S1[r] & S1[s] holds
( S1[r '&' s] & S1[r => s] )
proof
let r, s be Element of HP-WFF ; :: thesis: ( S1[r] & S1[s] implies ( S1[r '&' s] & S1[r => s] ) )
ex p9, q9 being DecoratedTree of HP-WFF st
( p9 = HP-Subformulae . r & q9 = HP-Subformulae . s & HP-Subformulae . (r '&' s) = (r '&' s) -tree (p9,q9) & HP-Subformulae . (r => s) = (r => s) -tree (p9,q9) ) by Def9;
hence ( S1[r] & S1[s] implies ( S1[r '&' s] & S1[r => s] ) ) ; :: thesis: verum
end;
A3: S1[ VERUM ] by Def9;
for p being Element of HP-WFF holds S1[p] from HILBERT2:sch 2(A3, A1, A2);
 hence HP-Subformulae . p is DecoratedTree of HP-WFF ; :: thesis: verum