let S be non void Signature; :: thesis: for X being V8() ManySortedSet of the carrier of S
for t being Term of S,X holds S variables_in t c= X

let X be V8() ManySortedSet of the carrier of S; :: thesis: for t being Term of S,X holds S variables_in t c= X
defpred S1[ DecoratedTree] means S variables_in \$1 c= X;
A1: for o being OperSymbol of S
for p being ArgumentSeq of Sym (o,X) st ( for t being Term of S,X st t in rng p holds
S1[t] ) holds
S1[[o, the carrier of S] -tree p]
proof
let o be OperSymbol of S; :: thesis: for p being ArgumentSeq of Sym (o,X) st ( for t being Term of S,X st t in rng p holds
S1[t] ) holds
S1[[o, the carrier of S] -tree p]

let p be ArgumentSeq of Sym (o,X); :: thesis: ( ( for t being Term of S,X st t in rng p holds
S1[t] ) implies S1[[o, the carrier of S] -tree p] )

assume A2: for t being Term of S,X st t in rng p holds
S variables_in t c= X ; :: thesis: S1[[o, the carrier of S] -tree p]
set q = [o, the carrier of S] -tree p;
thus S variables_in ([o, the carrier of S] -tree p) c= X :: thesis: verum
proof
let s9 be object ; :: according to PBOOLE:def 2 :: thesis: ( not s9 in the carrier of S or (S variables_in ([o, the carrier of S] -tree p)) . s9 c= X . s9 )
assume s9 in the carrier of S ; :: thesis: (S variables_in ([o, the carrier of S] -tree p)) . s9 c= X . s9
then reconsider z = s9 as SortSymbol of S ;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (S variables_in ([o, the carrier of S] -tree p)) . s9 or x in X . s9 )
assume x in (S variables_in ([o, the carrier of S] -tree p)) . s9 ; :: thesis: x in X . s9
then consider t being DecoratedTree such that
A3: t in rng p and
A4: x in () . z by Th11;
consider i being object such that
A5: i in dom p and
A6: t = p . i by ;
reconsider i = i as Nat by A5;
reconsider t = p . i as Term of S,X by ;
S variables_in t c= X by A2, A3, A6;
then (S variables_in t) . z c= X . z ;
hence x in X . s9 by A4, A6; :: thesis: verum
end;
end;
A7: for s being SortSymbol of S
for v being Element of X . s holds S1[ root-tree [v,s]]
proof
let s be SortSymbol of S; :: thesis: for v being Element of X . s holds S1[ root-tree [v,s]]
let v be Element of X . s; :: thesis: S1[ root-tree [v,s]]
thus S variables_in (root-tree [v,s]) c= X :: thesis: verum
proof
let s9 be object ; :: according to PBOOLE:def 2 :: thesis: ( not s9 in the carrier of S or (S variables_in (root-tree [v,s])) . s9 c= X . s9 )
assume s9 in the carrier of S ; :: thesis: (S variables_in (root-tree [v,s])) . s9 c= X . s9
then reconsider z = s9 as SortSymbol of S ;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (S variables_in (root-tree [v,s])) . s9 or x in X . s9 )
assume A8: x in (S variables_in (root-tree [v,s])) . s9 ; :: thesis: x in X . s9
then A9: ( s <> z implies x in {} ) by Th10;
( s = z implies x in {v} ) by ;
hence x in X . s9 by A9; :: thesis: verum
end;
end;
for t being Term of S,X holds S1[t] from hence for t being Term of S,X holds S variables_in t c= X ; :: thesis: verum