thus not tau1 . p is empty by Th6; :: thesis: tau1 . p is finite
defpred S₁[ Element of LTLB_WFF ] means tau1 . p is finite ;
A1: for n being Element of NAT holds S₁[ prop n]

proof

let n be Element of NAT ; :: thesis: S₁[ prop n]
tau1 . (prop n) = {(prop n)} by Def4;
hence S₁[ prop n] ; :: thesis: verum

end;

A2: for r, s being Element of LTLB_WFF st S₁[r] & S₁[s] holds
( S₁[r 'U' s] & S₁[r => s] )

proof

let r, s be Element of LTLB_WFF ; :: thesis: ( S₁[r] & S₁[s] implies ( S₁[r 'U' s] & S₁[r => s] ) )
assume A3: ( S₁[r] & S₁[s] ) ; :: thesis: ( S₁[r 'U' s] & S₁[r => s] )
tau1 . (r 'U' s) = {(r 'U' s)} by Def4;
hence S₁[r 'U' s] ; :: thesis: S₁[r => s]
tau1 . (r => s) = ({(r => s)} \/ (tau1 . r)) \/ (tau1 . s) by Def4;
hence S₁[r => s] by A3; :: thesis: verum

end;

A4: S₁[ TFALSUM ] by Def4;
for p being Element of LTLB_WFF holds S₁[p] from HILBERT2:sch 2(A4, A1, A2);
hence tau1 . p is finite ; :: thesis: verum