let p be Element of LTLB_WFF ; :: thesis:
defpred S1[ Element of LTLB_WFF ] means tau1 . \$1 c= Sub . \$1;
set f = TFALSUM ;
A1: for n being Element of NAT holds S1[ prop n]
proof
let n be Element of NAT ; :: thesis: S1[ prop n]
set pr = prop n;
tau1 . (prop n) = {(prop n)} by Def4
.= Sub . (prop n) by Def6 ;
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 that
A3: S1[r] and
A4: S1[s] ; :: thesis: ( S1[r 'U' s] & S1[r => s] )
thus S1[r 'U' s] :: thesis: S1[r => s]
proof
set f = r 'U' s;
{(r 'U' s)} c= Sub . (r 'U' s) by ;
hence S1[r 'U' s] by Def4; :: thesis: verum
end;
thus S1[r => s] :: thesis: verum
proof
set f = r => s;
A5: Sub . (r => s) = ({(r => s)} \/ (Sub . r)) \/ (Sub . s) by Def6;
( {(r => s)} \/ () c= {(r => s)} \/ (Sub . r) & tau1 . (r => s) = ({(r => s)} \/ ()) \/ () ) by ;
hence S1[r => s] by ; :: thesis: verum
end;
end;
tau1 . TFALSUM = by Def4
.= Sub . TFALSUM by Def6 ;
then A6: S1[ TFALSUM ] ;
for p being Element of LTLB_WFF holds S1[p] from HILBERT2:sch 2(A6, A1, A2);
hence tau1 . p c= Sub . p ; :: thesis: verum