reserve A,B,p,q,r,s for Element of LTLB_WFF,
  n for Element of NAT,
  X for Subset of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y for set;

theorem p in Sub.(A 'U' B) implies (A 'U' B in Sub.q implies p in Sub.q)
  proof
    set aub = A 'U' B,f = TFALSUM;
    defpred P1[Element of l] means aub in Sub.$1 implies p in Sub.$1;
A1: for n holds P1[ prop n]
    proof
      let n;
      set pr = prop n;
      assume aub in Sub.pr;
      then aub in {pr} by Def6;
      then aub = pr by TARSKI:def 1;
      hence thesis by HILBERT2:24;
    end;
    assume
A2: p in Sub.aub;
A3: for r, s st P1[r] & P1[s] holds P1[r 'U' s] & P1[r => s]
    proof
      let r,s;
      assume that
A4:   P1[r] and
A5:   P1[s];
      thus P1[r 'U' s]
      proof
        set f = r 'U' s;
A6:     tau1.r c= Sub.r by Th25;
A7:     Sub. f = tau1.untn(r,s) \/ Sub.r \/ Sub.s by Def6;
        then A8: Sub.s c= Sub.f by XBOOLE_1:7;
        Sub.r c= tau1.untn(r,s) \/ Sub.r & tau1.untn(r,s) \/ Sub.r c= Sub.f
        by XBOOLE_1:7, A7;
        then A9: Sub.r c= Sub.f;
        assume aub in Sub.f;
        then A10: aub in tau1.untn(r,s) \/ Sub.r \/ Sub.s by Def6;
A11:    tau1.s c= Sub.s by Th25;
        per cases by A10,XBOOLE_0:def 3;
        suppose
A12:      aub in tau1.untn(r,s) \/ Sub.r;
          per cases by A12,XBOOLE_0:def 3;
          suppose
            aub in tau1.untn(r,s);then
            aub in tau1.(('not' s) '&&' ('not' (r '&&' f))) by Th9;then
            aub in tau1.('not' s) or aub in tau1.('not' (r '&&' f)) by Th10;
            then A13: aub in tau1.s or aub in tau1.(r '&&' f) by Th9;
            per cases by A13,Th10;
            suppose
A14:          aub in tau1.r or aub in tau1.s;
              per cases by A14;
              suppose aub in tau1.r;
                hence p in Sub.f by A6,A4,A9;
              end;
              suppose aub in tau1.s;
                hence p in Sub.f by A11,A5,A8;
              end;
            end;
            suppose aub in tau1.f;
              then aub in {f} by Def4;
              hence p in Sub.f by TARSKI:def 1,A2;
            end;
          end;
          suppose aub in Sub.r;
            hence p in Sub.f by A4,A9;
          end;
        end;
        suppose aub in Sub.s;
          hence p in Sub.f by A5,A8;
        end;
      end;
      thus P1[r => s]
      proof
        set f = r => s;
A15:    Sub.f = {f} \/ Sub.r \/ Sub.s by Def6;
        then A16: Sub.s c= Sub.f by XBOOLE_1:7;
        Sub.r c= {f} \/ Sub.r & {f} \/ Sub.r c= Sub.f by XBOOLE_1:7, A15;
        then A17: Sub.r c= Sub.f;
        assume aub in Sub.f;
        then A18: aub in {f} \/ Sub.r \/ Sub.s by Def6;
        per cases by A18,XBOOLE_0:def 3;
        suppose
A19:      aub in {f} \/ Sub.r;
          per cases by A19,XBOOLE_0:def 3;
          suppose aub in {f};
            then aub = f by TARSKI:def 1;
            hence p in Sub.f by HILBERT2:22;
          end;
          suppose aub in Sub.r;
            hence p in Sub.f by A4,A17;
          end;
        end;
        suppose aub in Sub.s;
          hence p in Sub.f by A5,A16;
        end;
      end;
    end;
A20: P1[f]
     proof
       assume aub in Sub.f;
       then aub in {f} by Def6;
       then aub = f by TARSKI:def 1;
       hence thesis by HILBERT2:23;
     end;
     for p holds P1[p] from HILBERT2:sch 2(A20,A1,A3);
     hence thesis;
   end;
