:: BOOLE  semantic presentation begin  
theorem :: BOOLE:1 (Bool  "for" (Set (Var "X")) "being"    ($#m1_hidden :::"set"::: )   "holds"  (Bool (Set (Set (Var "X"))  ($#k2_xboole_0 :::"\/"::: )  (Set  ($#k1_xboole_0 :::"{}"::: ) ))  ($#r1_hidden :::"="::: )  (Set (Var "X")))) ; 
 
theorem :: BOOLE:2 (Bool  "for" (Set (Var "X")) "being"    ($#m1_hidden :::"set"::: )   "holds"  (Bool (Set (Set (Var "X"))  ($#k3_xboole_0 :::"/\"::: )  (Set  ($#k1_xboole_0 :::"{}"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k1_xboole_0 :::"{}"::: )  ))) ; 
 
theorem :: BOOLE:3 (Bool  "for" (Set (Var "X")) "being"    ($#m1_hidden :::"set"::: )   "holds"  (Bool (Set (Set (Var "X"))  ($#k4_xboole_0 :::"\"::: )  (Set  ($#k1_xboole_0 :::"{}"::: ) ))  ($#r1_hidden :::"="::: )  (Set (Var "X")))) ; 
 
theorem :: BOOLE:4 (Bool  "for" (Set (Var "X")) "being"    ($#m1_hidden :::"set"::: )   "holds"  (Bool (Set (Set  ($#k1_xboole_0 :::"{}"::: ) )  ($#k4_xboole_0 :::"\"::: )  (Set (Var "X")))  ($#r1_hidden :::"="::: )  (Set   ($#k1_xboole_0 :::"{}"::: )  ))) ; 
 
theorem :: BOOLE:5 (Bool  "for" (Set (Var "X")) "being"    ($#m1_hidden :::"set"::: )   "holds"  (Bool (Set (Set (Var "X"))  ($#k5_xboole_0 :::"\+\"::: )  (Set  ($#k1_xboole_0 :::"{}"::: ) ))  ($#r1_hidden :::"="::: )  (Set (Var "X")))) ; 
 




theorem :: BOOLE:6 (Bool  "for" (Set (Var "X")) "being"    ($#m1_hidden :::"set"::: )    "st" (Bool (Bool (Set (Var "X")) "is"   ($#v1_xboole_0 :::"empty"::: )  )) "holds"  (Bool (Set (Var "X"))  ($#r1_hidden :::"="::: )  (Set   ($#k1_xboole_0 :::"{}"::: )  ))) ; 
 
theorem :: BOOLE:7 (Bool  "for" (Set (Var "x")) "," (Set (Var "X")) "being"    ($#m1_hidden :::"set"::: )    "st" (Bool (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set (Var "X")))) "holds"  (Bool  "not" (Bool (Set (Var "X")) "is"   ($#v1_xboole_0 :::"empty"::: )  ))) ; 
 
theorem :: BOOLE:8 (Bool  "for" (Set (Var "X")) "," (Set (Var "Y")) "being"    ($#m1_hidden :::"set"::: )    "st" (Bool (Bool (Set (Var "X")) "is"   ($#v1_xboole_0 :::"empty"::: )  ) & (Bool (Set (Var "X"))  ($#r1_hidden :::"<>"::: )  (Set (Var "Y")))) "holds"  (Bool  "not" (Bool (Set (Var "Y")) "is"   ($#v1_xboole_0 :::"empty"::: )  ))) ;