:: BCIALG_3 semantic presentation begin definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) ; attr "IT" is :::"commutative"::: means :: BCIALG_3:def 1 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))); end; :: deftheorem defines :::"commutative"::: BCIALG_3:def 1 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))) ")" )); registration cluster (Set ($#k3_bcialg_1 :::"BCI-EXAMPLE"::: ) ) -> ($#v1_bcialg_3 :::"commutative"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_bcialg_1 :::"being_B"::: ) ($#v4_bcialg_1 :::"being_C"::: ) ($#v5_bcialg_1 :::"being_I"::: ) ($#v7_bcialg_1 :::"being_BCI-4"::: ) ($#v8_bcialg_1 :::"being_BCK-5"::: ) ($#v1_bcialg_3 :::"commutative"::: ) for ($#l2_bcialg_1 :::"BCIStr_0"::: ) ; end; theorem :: BCIALG_3:1 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_bcialg_1 :::"<="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))) ")" )) ; theorem :: BCIALG_3:2 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "y"))) & (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "x"))) ")" ))) ; theorem :: BCIALG_3:3 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" )))) ")" )) ; theorem :: BCIALG_3:4 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ")" )))) ")" )) ; theorem :: BCIALG_3:5 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "y")))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))) ")" )) ; theorem :: BCIALG_3:6 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "X")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ))) ")" )) ")" )) ; theorem :: BCIALG_3:7 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))) "holds" (Bool (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "y"))))) ; theorem :: BCIALG_3:8 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "y")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "a")))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")))))) ; theorem :: BCIALG_3:9 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) "iff" (Bool (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X")))) ")" ))) ; theorem :: BCIALG_3:10 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")))) & (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")))) ")" ))) ; theorem :: BCIALG_3:11 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "a")))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ))))) ; definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "X")); attr "a" is :::"being_greatest"::: means :: BCIALG_3:def 2 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "X" "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) "a") ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "X"))); attr "a" is :::"being_positive"::: means :: BCIALG_3:def 3 (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) "X" ")" ) ($#k1_bcialg_1 :::"\"::: ) "a") ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "X")); end; :: deftheorem defines :::"being_greatest"::: BCIALG_3:def 2 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X"))))) ")" ))); :: deftheorem defines :::"being_positive"::: BCIALG_3:def 3 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#v3_bcialg_3 :::"being_positive"::: ) ) "iff" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "X")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X")))) ")" ))); begin definitionlet "IT" be ($#l2_bcialg_1 :::"BCI-algebra":::); attr "IT" is :::"BCI-commutative"::: means :: BCIALG_3:def 4 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "st" (Bool (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "IT"))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))); attr "IT" is :::"BCI-weakly-commutative"::: means :: BCIALG_3:def 5 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" ($#k4_struct_0 :::"0."::: ) "IT" ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))); end; :: deftheorem defines :::"BCI-commutative"::: BCIALG_3:def 4 : (Bool "for" (Set (Var "IT")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "st" (Bool (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "IT"))))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))) ")" )); :: deftheorem defines :::"BCI-weakly-commutative"::: BCIALG_3:def 5 : (Bool "for" (Set (Var "IT")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v5_bcialg_3 :::"BCI-weakly-commutative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "IT")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))) ")" )); registration cluster (Set ($#k3_bcialg_1 :::"BCI-EXAMPLE"::: ) ) -> ($#v4_bcialg_3 :::"BCI-commutative"::: ) ($#v5_bcialg_3 :::"BCI-weakly-commutative"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_bcialg_1 :::"being_B"::: ) ($#v4_bcialg_1 :::"being_C"::: ) ($#v5_bcialg_1 :::"being_I"::: ) ($#v7_bcialg_1 :::"being_BCI-4"::: ) ($#v4_bcialg_3 :::"BCI-commutative"::: ) ($#v5_bcialg_3 :::"BCI-weakly-commutative"::: ) for ($#l2_bcialg_1 :::"BCIStr_0"::: ) ; end; theorem :: BCIALG_3:12 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "st" (Bool (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) ))) "holds" (Bool (Set (Var "X")) "is" ($#l2_bcialg_1 :::"BCK-algebra":::))) ; theorem :: BCIALG_3:13 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v19_bcialg_1 :::"p-Semisimple"::: ) )) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ) & (Bool (Set (Var "X")) "is" ($#v5_bcialg_3 :::"BCI-weakly-commutative"::: ) ) ")" )) ; theorem :: BCIALG_3:14 (Bool "for" (Set (Var "X")) "being" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ($#l2_bcialg_1 :::"BCI-algebra":::)) & (Bool (Set (Var "X")) "is" ($#v5_bcialg_3 :::"BCI-weakly-commutative"::: ) ($#l2_bcialg_1 :::"BCI-algebra":::)) ")" )) ; theorem :: BCIALG_3:15 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v5_bcialg_3 :::"BCI-weakly-commutative"::: ) ($#l2_bcialg_1 :::"BCI-algebra":::))) "holds" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) )) ; theorem :: BCIALG_3:16 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ")" )))) ")" )) ; theorem :: BCIALG_3:17 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "X")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )))) ")" )) ; theorem :: BCIALG_3:18 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "a"))) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ))))) ")" )) ; theorem :: BCIALG_3:19 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ($#l2_bcialg_1 :::"BCI-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "z")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X")))) & (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ")" ))) ")" )) ")" )) ; theorem :: BCIALG_3:20 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "z"))) & (Bool (Set (Set (Var "z")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_bcialg_1 :::"<="::: ) (Set (Set (Var "z")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x"))))) "holds" (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "y")))) ")" )) ; theorem :: BCIALG_3:21 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_bcialg_3 :::"BCI-commutative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "y"))) & (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "z")))) "holds" (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" )))) ")" )) ; begin definitionlet "IT" be ($#l2_bcialg_1 :::"BCK-algebra":::); attr "IT" is :::"bounded"::: means :: BCIALG_3:def 6 (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "st" (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )); end; :: deftheorem defines :::"bounded"::: BCIALG_3:def 6 : (Bool "for" (Set (Var "IT")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v6_bcialg_3 :::"bounded"::: ) ) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "st" (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) ")" )); registration cluster (Set ($#k3_bcialg_1 :::"BCI-EXAMPLE"::: ) ) -> ($#v6_bcialg_3 :::"bounded"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_bcialg_1 :::"being_B"::: ) ($#v4_bcialg_1 :::"being_C"::: ) ($#v5_bcialg_1 :::"being_I"::: ) ($#v7_bcialg_1 :::"being_BCI-4"::: ) ($#v8_bcialg_1 :::"being_BCK-5"::: ) ($#v1_bcialg_3 :::"commutative"::: ) ($#v6_bcialg_3 :::"bounded"::: ) for ($#l2_bcialg_1 :::"BCIStr_0"::: ) ; end; definitionlet "IT" be ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::); attr "IT" is :::"involutory"::: means :: BCIALG_3:def 7 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))))); end; :: deftheorem defines :::"involutory"::: BCIALG_3:def 7 : (Bool "for" (Set (Var "IT")) "being" ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v7_bcialg_3 :::"involutory"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ")" )); theorem :: BCIALG_3:22 (Bool "for" (Set (Var "X")) "being" ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v7_bcialg_3 :::"involutory"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ))))) ")" )) ; theorem :: BCIALG_3:23 (Bool "for" (Set (Var "X")) "being" ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v7_bcialg_3 :::"involutory"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ))))) ")" )) ; theorem :: BCIALG_3:24 (Bool "for" (Set (Var "X")) "being" ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v7_bcialg_3 :::"involutory"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))))) "holds" (Bool (Set (Var "y")) ($#r1_bcialg_1 :::"<="::: ) (Set (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")))))) ")" )) ; definitionlet "IT" be ($#l2_bcialg_1 :::"BCK-algebra":::); let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "IT")); attr "a" is :::"being_Iseki"::: means :: BCIALG_3:def 8 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) "a") ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "IT")) & (Bool (Set "a" ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) "a") ")" )); end; :: deftheorem defines :::"being_Iseki"::: BCIALG_3:def 8 : (Bool "for" (Set (Var "IT")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#v8_bcialg_3 :::"being_Iseki"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "IT")))) & (Bool (Set (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" )) ")" ))); definitionlet "IT" be ($#l2_bcialg_1 :::"BCK-algebra":::); attr "IT" is :::"Iseki_extension"::: means :: BCIALG_3:def 9 (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "st" (Bool (Set (Var "a")) "is" ($#v8_bcialg_3 :::"being_Iseki"::: ) )); end; :: deftheorem defines :::"Iseki_extension"::: BCIALG_3:def 9 : (Bool "for" (Set (Var "IT")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v9_bcialg_3 :::"Iseki_extension"::: ) ) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "st" (Bool (Set (Var "a")) "is" ($#v8_bcialg_3 :::"being_Iseki"::: ) )) ")" )); registration cluster (Set ($#k3_bcialg_1 :::"BCI-EXAMPLE"::: ) ) -> ($#v9_bcialg_3 :::"Iseki_extension"::: ) ; end; definitionlet "X" be ($#l2_bcialg_1 :::"BCK-algebra":::); mode :::"Commutative-Ideal"::: "of" "X" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" "X" means :: BCIALG_3:def 10 (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) "X") ($#r2_hidden :::"in"::: ) it) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" "X" "st" (Bool (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z"))) ($#r2_hidden :::"in"::: ) it) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" )) ($#r2_hidden :::"in"::: ) it) ")" ) ")" ); end; :: deftheorem defines :::"Commutative-Ideal"::: BCIALG_3:def 10 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "b2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_bcialg_3 :::"Commutative-Ideal"::: ) "of" (Set (Var "X"))) "iff" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z"))) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "b2")))) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) ")" ) ")" ) ")" ))); theorem :: BCIALG_3:25 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "IT")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "IT")) "is" ($#m1_bcialg_3 :::"Commutative-Ideal"::: ) "of" (Set (Var "X")))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))))) ; theorem :: BCIALG_3:26 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "IT")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "st" (Bool (Bool (Set (Var "IT")) "is" ($#m1_bcialg_3 :::"Commutative-Ideal"::: ) "of" (Set (Var "X")))) "holds" (Bool (Set (Var "IT")) "is" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")))))) ; theorem :: BCIALG_3:27 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "IT")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "IT")) "is" ($#m1_bcialg_3 :::"Commutative-Ideal"::: ) "of" (Set (Var "X")))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))) "holds" (Bool (Set (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))))) ; begin definitionlet "IT" be ($#l2_bcialg_1 :::"BCK-algebra":::); attr "IT" is :::"BCK-positive-implicative"::: means :: BCIALG_3:def 11 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" )))); attr "IT" is :::"BCK-implicative"::: means :: BCIALG_3:def 12 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x")))); end; :: deftheorem defines :::"BCK-positive-implicative"::: BCIALG_3:def 11 : (Bool "for" (Set (Var "IT")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" )))) ")" )); :: deftheorem defines :::"BCK-implicative"::: BCIALG_3:def 12 : (Bool "for" (Set (Var "IT")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x")))) ")" )); registration cluster (Set ($#k3_bcialg_1 :::"BCI-EXAMPLE"::: ) ) -> ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#v11_bcialg_3 :::"BCK-implicative"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_bcialg_1 :::"being_B"::: ) ($#v4_bcialg_1 :::"being_C"::: ) ($#v5_bcialg_1 :::"being_I"::: ) ($#v7_bcialg_1 :::"being_BCI-4"::: ) ($#v8_bcialg_1 :::"being_BCK-5"::: ) ($#v1_bcialg_3 :::"commutative"::: ) ($#v6_bcialg_3 :::"bounded"::: ) ($#v9_bcialg_3 :::"Iseki_extension"::: ) ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#v11_bcialg_3 :::"BCK-implicative"::: ) for ($#l2_bcialg_1 :::"BCIStr_0"::: ) ; end; theorem :: BCIALG_3:28 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))))) ")" )) ; theorem :: BCIALG_3:29 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" ) ")" )))) ")" )) ; theorem :: BCIALG_3:30 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" )))) ")" )) ; theorem :: BCIALG_3:31 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" )) ($#r1_bcialg_1 :::"<="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z"))))) ")" )) ; theorem :: BCIALG_3:32 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_bcialg_1 :::"<="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))))) ")" )) ; theorem :: BCIALG_3:33 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" ) ")" )) ($#r1_bcialg_1 :::"<="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))) ")" )) ; theorem :: BCIALG_3:34 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_bcialg_3 :::"commutative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) & (Bool (Set (Var "X")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) ")" ) ")" )) ; theorem :: BCIALG_3:35 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )))) ")" )) ; theorem :: BCIALG_3:36 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "X")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ))) ")" )) ")" )) ; theorem :: BCIALG_3:37 (Bool "for" (Set (Var "X")) "being" ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ) "iff" (Bool "(" (Bool (Set (Var "X")) "is" ($#v7_bcialg_3 :::"involutory"::: ) ) & (Bool (Set (Var "X")) "is" ($#v10_bcialg_3 :::"BCK-positive-implicative"::: ) ) ")" ) ")" ))) ; theorem :: BCIALG_3:38 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X"))))) ")" )) ; theorem :: BCIALG_3:39 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ")" ) ")" )))) ")" )) ; theorem :: BCIALG_3:40 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" )))) ")" )) ; theorem :: BCIALG_3:41 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ")" )))) ")" )) ; theorem :: BCIALG_3:42 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCK-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )))) ")" )) ; theorem :: BCIALG_3:43 (Bool "for" (Set (Var "X")) "being" ($#v1_bcialg_3 :::"commutative"::: ) ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X"))))) ")" ))) ; theorem :: BCIALG_3:44 (Bool "for" (Set (Var "X")) "being" ($#v1_bcialg_3 :::"commutative"::: ) ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x")))) ")" ))) ; theorem :: BCIALG_3:45 (Bool "for" (Set (Var "X")) "being" ($#v1_bcialg_3 :::"commutative"::: ) ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x"))))) ")" ))) ; theorem :: BCIALG_3:46 (Bool "for" (Set (Var "X")) "being" ($#v1_bcialg_3 :::"commutative"::: ) ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))))) ")" ))) ; theorem :: BCIALG_3:47 (Bool "for" (Set (Var "X")) "being" ($#v1_bcialg_3 :::"commutative"::: ) ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )))) ")" ))) ; theorem :: BCIALG_3:48 (Bool "for" (Set (Var "X")) "being" ($#v1_bcialg_3 :::"commutative"::: ) ($#v6_bcialg_3 :::"bounded"::: ) ($#l2_bcialg_1 :::"BCK-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_bcialg_3 :::"being_greatest"::: ) )) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v11_bcialg_3 :::"BCK-implicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "y")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_bcialg_1 :::"<="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "z")) ")" )))) ")" ))) ;