let S be non void Signature; :: thesis: for X being non-empty ManySortedSet of the carrier of S
for t being Term of S,X holds S variables_in t c= X
let X be non-empty 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 (S variables_in t) . z by Th11;
consider i being object such that
A5: i in dom p and
A6: t = p . i by A3, FUNCT_1:def 3;
reconsider i = i as Nat by A5;
reconsider t = p . i as Term of S,X by A5, MSATERM:22;
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 A8, Th10;
hence x in X . s9 by A9; :: thesis: verum

end;

end;

for t being Term of S,X holds S₁[t] from MSATERM:sch 1(A7, A1);
hence for t being Term of S,X holds S variables_in t c= X ; :: thesis: verum