let p be Element of LTLB_WFF ; :: thesis: p in tau1 . p

defpred S_{1}[ Element of LTLB_WFF ] means $1 in tau1 . $1;

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] )

then A3: S_{1}[ TFALSUM ]
by TARSKI:def 1;

for p being Element of LTLB_WFF holds S_{1}[p]
from HILBERT2:sch 2(A3, A1, A2);

hence p in tau1 . p ; :: 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]
by TARSKI:def 1; :: thesis: verum

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

hence S

( S

proof

tau1 . TFALSUM = {TFALSUM}
by Def4;
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 that

S_{1}[r]
and

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]
by TARSKI:def 1; :: thesis: S_{1}[r => s]

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

then ( {(r => s)} c= {(r => s)} \/ (tau1 . r) & {(r => s)} \/ (tau1 . r) c= tau1 . (r => s) ) by XBOOLE_1:7;

hence S_{1}[r => s]
by ZFMISC_1:31; :: thesis: verum

end;assume that

S

S

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

hence S

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

then ( {(r => s)} c= {(r => s)} \/ (tau1 . r) & {(r => s)} \/ (tau1 . r) c= tau1 . (r => s) ) by XBOOLE_1:7;

hence S

then A3: S

for p being Element of LTLB_WFF holds S

hence p in tau1 . p ; :: thesis: verum