thus
not tau1 . p is empty
by Th6; :: thesis: tau1 . p is finite

defpred S_{1}[ Element of LTLB_WFF ] means tau1 . p is finite ;

A1: for n being Element of NAT holds S_{1}[ prop n]
_{1}[r] & S_{1}[s] holds

( S_{1}[r 'U' s] & S_{1}[r => s] )
_{1}[ TFALSUM ]
by Def4;

for p being Element of LTLB_WFF holds S_{1}[p]
from HILBERT2:sch 2(A4, A1, A2);

hence tau1 . p is finite ; :: thesis: verum

defpred S

A1: for n being Element of NAT holds S

proof

A2:
for r, s being Element of LTLB_WFF st S
let n be Element of NAT ; :: thesis: S_{1}[ prop n]

tau1 . (prop n) = {(prop n)} by Def4;

hence S_{1}[ prop n]
; :: thesis: verum

end;tau1 . (prop n) = {(prop n)} by Def4;

hence S

( S

proof

A4:
S
let r, s be Element of LTLB_WFF ; :: thesis: ( S_{1}[r] & S_{1}[s] implies ( S_{1}[r 'U' s] & S_{1}[r => s] ) )

assume A3: ( S_{1}[r] & S_{1}[s] )
; :: thesis: ( S_{1}[r 'U' s] & S_{1}[r => s] )

tau1 . (r 'U' s) = {(r 'U' s)} by Def4;

hence S_{1}[r 'U' s]
; :: thesis: S_{1}[r => s]

tau1 . (r => s) = ({(r => s)} \/ (tau1 . r)) \/ (tau1 . s) by Def4;

hence S_{1}[r => s]
by A3; :: thesis: verum

end;assume A3: ( S

tau1 . (r 'U' s) = {(r 'U' s)} by Def4;

hence S

tau1 . (r => s) = ({(r => s)} \/ (tau1 . r)) \/ (tau1 . s) by Def4;

hence S

for p being Element of LTLB_WFF holds S

hence tau1 . p is finite ; :: thesis: verum