:: BCIALG_6 semantic presentation begin definitionlet "D" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set (Const "D")); let "n" be ($#m1_hidden :::"Nat":::); :: original: :::"."::: redefine func "f" :::"."::: "n" -> ($#m1_subset_1 :::"Element"::: ) "of" "D"; end; definitionlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) ; func :::"BCI-power"::: "G" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "G") "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "G") means :: BCIALG_6:def 1 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "G" "holds" (Bool "(" (Bool (Set it ($#k2_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "G")) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k2_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" it ($#k2_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "n")) ")" ")" ) ($#k2_bcialg_1 :::"`"::: ) ")" ))) ")" ) ")" )); end; :: deftheorem defines :::"BCI-power"::: BCIALG_6:def 1 : (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "G"))) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "G"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_bcialg_6 :::"BCI-power"::: ) (Set (Var "G")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Set (Var "b2")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "G")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b2")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "b2")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "n")) ")" ")" ) ($#k2_bcialg_1 :::"`"::: ) ")" ))) ")" ) ")" )) ")" ))); definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "i" be ($#m1_hidden :::"Integer":::); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "X")); func "x" :::"|^"::: "i" -> ($#m1_subset_1 :::"Element":::) "of" "X" equals :: BCIALG_6:def 2 (Set (Set "(" ($#k2_bcialg_6 :::"BCI-power"::: ) "X" ")" ) ($#k2_binop_1 :::"."::: ) "(" "x" "," (Set "(" ($#k1_int_2 :::"abs"::: ) "i" ")" ) ")" ) if (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) "i") otherwise (Set (Set "(" ($#k2_bcialg_6 :::"BCI-power"::: ) "X" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set "(" "x" ($#k2_bcialg_1 :::"`"::: ) ")" ) "," (Set "(" ($#k1_int_2 :::"abs"::: ) "i" ")" ) ")" ); end; :: deftheorem defines :::"|^"::: BCIALG_6:def 2 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_bcialg_6 :::"BCI-power"::: ) (Set (Var "X")) ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set "(" ($#k1_int_2 :::"abs"::: ) (Set (Var "i")) ")" ) ")" )) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))))) "implies" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_bcialg_6 :::"BCI-power"::: ) (Set (Var "X")) ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ")" ) "," (Set "(" ($#k1_int_2 :::"abs"::: ) (Set (Var "i")) ")" ) ")" )) ")" ")" )))); definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "n" be ($#m1_hidden :::"Nat":::); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "X")); redefine func "x" :::"|^"::: "n" equals :: BCIALG_6:def 3 (Set (Set "(" ($#k2_bcialg_6 :::"BCI-power"::: ) "X" ")" ) ($#k1_binop_1 :::"."::: ) "(" "x" "," "n" ")" ); end; :: deftheorem defines :::"|^"::: BCIALG_6:def 3 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_bcialg_6 :::"BCI-power"::: ) (Set (Var "X")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "n")) ")" ))))); theorem :: BCIALG_6:1 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) "holds" (Bool (Set (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "a")) ")" )))))) ; theorem :: BCIALG_6:2 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k2_bcialg_1 :::"`"::: ) ")" )))))) ; theorem :: BCIALG_6:3 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X")))))) ; theorem :: BCIALG_6:4 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: BCIALG_6:5 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) )))) ; theorem :: BCIALG_6:6 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ")" ))))) ; theorem :: BCIALG_6:7 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "X")) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X")))))) ; theorem :: BCIALG_6:8 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "a"))))) ; theorem :: BCIALG_6:9 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" )))))) ; theorem :: BCIALG_6:10 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" )))))) ; theorem :: BCIALG_6:11 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) (Set (Var "X")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))))) ; theorem :: BCIALG_6:12 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X"))))))) ; theorem :: BCIALG_6:13 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))))))) ; theorem :: BCIALG_6:14 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k2_bcialg_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "a"))))))) ; theorem :: BCIALG_6:15 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k6_bcialg_1 :::"\"::: ) (Set (Var "b")) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "b")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n")) ")" )))))) ; theorem :: BCIALG_6:16 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k6_bcialg_1 :::"\"::: ) (Set (Var "b")) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "b")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" ) ")" )))))) ; theorem :: BCIALG_6:17 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k2_bcialg_1 :::"`"::: ) ))))) ; theorem :: BCIALG_6:18 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k2_bcialg_1 :::"`"::: ) ))))) ; theorem :: BCIALG_6:19 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ($#k2_bcialg_1 :::"`"::: ) ))))) ; theorem :: BCIALG_6:20 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "a")))))))) ; theorem :: BCIALG_6:21 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k2_bcialg_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k2_bcialg_1 :::"`"::: ) ))))) ; theorem :: BCIALG_6:22 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "i")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "j")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" (Set (Var "i")) ($#k21_binop_2 :::"-"::: ) (Set (Var "j")) ")" )))))) ; theorem :: BCIALG_6:23 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "i")) ")" ) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" (Set (Var "i")) ($#k22_binop_2 :::"*"::: ) (Set (Var "j")) ")" )))))) ; theorem :: BCIALG_6:24 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set "(" (Set (Var "i")) ($#k20_binop_2 :::"+"::: ) (Set (Var "j")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "i")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "j")) ")" ) ($#k2_bcialg_1 :::"`"::: ) ")" )))))) ; definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "X")); attr "x" is :::"finite-period"::: means :: BCIALG_6:def 4 (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set "x" ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) "X")) ")" )); end; :: deftheorem defines :::"finite-period"::: BCIALG_6:def 4 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) (Set (Var "X")))) ")" )) ")" ))); theorem :: BCIALG_6:25 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ")" ) ($#k2_bcialg_1 :::"`"::: ) ) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ))) ; definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "X")); assume (Bool (Set (Const "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) ; func :::"ord"::: "x" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: BCIALG_6:def 5 (Bool "(" (Bool (Set "x" ($#k3_bcialg_6 :::"|^"::: ) it) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) "X")) & (Bool it ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set "x" ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) "X")) & (Bool (Set (Var "m")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool it ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) ")" ) ")" ); end; :: deftheorem defines :::"ord"::: BCIALG_6:def 5 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "b3"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) (Set (Var "X")))) & (Bool (Set (Var "b3")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) (Set (Var "X")))) & (Bool (Set (Var "m")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "b3")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) ")" ) ")" ) ")" )))); theorem :: BCIALG_6:26 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "a")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "a")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X"))))))) ; theorem :: BCIALG_6:27 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#l2_bcialg_1 :::"BCK-algebra":::)) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) ")" )) ; theorem :: BCIALG_6:28 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Var "a")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "a"))))) "holds" (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "a"))))))) ; theorem :: BCIALG_6:29 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_bcialg_1 :::"BCK-part"::: ) (Set (Var "X")))) "iff" (Bool (Set (Var "n")) ($#r1_int_1 :::"divides"::: ) (Set (Var "m"))) ")" )))) ; theorem :: BCIALG_6:30 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "m"))) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set "(" (Set (Var "x")) ($#k3_bcialg_6 :::"|^"::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_int_1 :::"div"::: ) (Set "(" (Set (Var "m")) ($#k6_nat_d :::"gcd"::: ) (Set (Var "n")) ")" )))))) ; theorem :: BCIALG_6:31 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ) "is" ($#v1_bcialg_6 :::"finite-period"::: ) )) "holds" (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_bcialg_6 :::"ord"::: ) (Set "(" (Set (Var "x")) ($#k2_bcialg_1 :::"`"::: ) ")" ))))) ; theorem :: BCIALG_6:32 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) "st" (Bool (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "a")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "a"))))) "holds" (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set "(" (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))))) ; theorem :: BCIALG_6:33 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) "st" (Bool (Bool (Set (Set (Var "a")) ($#k6_bcialg_1 :::"\"::: ) (Set (Var "b"))) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Var "x")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Var "y")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Var "a")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Var "b")) "is" ($#v1_bcialg_6 :::"finite-period"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "a")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "b"))))) "holds" (Bool (Set ($#k4_bcialg_6 :::"ord"::: ) (Set "(" (Set (Var "a")) ($#k6_bcialg_1 :::"\"::: ) (Set (Var "b")) ")" )) ($#r1_int_1 :::"divides"::: ) (Set (Set "(" ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "x")) ")" ) ($#k5_nat_d :::"lcm"::: ) (Set "(" ($#k4_bcialg_6 :::"ord"::: ) (Set (Var "y")) ")" )))))) ; begin theorem :: BCIALG_6:34 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "Y")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "x9")) "," (Set (Var "y9")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "x9"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "y9")))) "holds" (Bool (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x9")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y9")))))))) ; definitionlet "X", "X9" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "X9")); attr "f" is :::"multiplicative"::: means :: BCIALG_6:def 6 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" "X" "holds" (Bool (Set "f" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "b")) ")" )))); end; :: deftheorem defines :::"multiplicative"::: BCIALG_6:def 6 : (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_bcialg_1 :::"BCIStr_0"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_bcialg_6 :::"multiplicative"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) ($#k1_bcialg_1 :::"\"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "b")) ")" )))) ")" ))); registrationlet "X", "X9" be ($#l2_bcialg_1 :::"BCI-algebra":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X9") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_PARTFUN1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X")) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_bcialg_6 :::"multiplicative"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X9") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "X", "X9" be ($#l2_bcialg_1 :::"BCI-algebra":::); mode BCI-homomorphism of "X" "," "X9" is ($#v2_bcialg_6 :::"multiplicative"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," "X9"; end; definitionlet "X", "X9" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "f" be ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Const "X")) "," (Set (Const "X9")); attr "f" is :::"isotonic"::: means :: BCIALG_6:def 7 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "X" "st" (Bool (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "y")))) "holds" (Bool (Set "f" ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_bcialg_1 :::"<="::: ) (Set "f" ($#k3_funct_2 :::"."::: ) (Set (Var "y"))))); end; :: deftheorem defines :::"isotonic"::: BCIALG_6:def 7 : (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_bcialg_6 :::"isotonic"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "y")))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_bcialg_1 :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y"))))) ")" ))); definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); mode Endomorphism of "X" is ($#m1_subset_1 :::"BCI-homomorphism":::) "of" "X" "," "X"; end; definitionlet "X", "X9" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "f" be ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Const "X")) "," (Set (Const "X9")); func :::"Ker"::: "f" -> ($#m1_hidden :::"set"::: ) equals :: BCIALG_6:def 8 "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" "X" : (Bool (Set "f" ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "X9")) "}" ; end; :: deftheorem defines :::"Ker"::: BCIALG_6:def 8 : (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "holds" (Bool (Set ($#k5_bcialg_6 :::"Ker"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) : (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X9")))) "}" ))); theorem :: BCIALG_6:35 (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X9")))))) ; registrationlet "X", "X9" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "f" be ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Const "X")) "," (Set (Const "X9")); cluster (Set ($#k5_bcialg_6 :::"Ker"::: ) "f") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: BCIALG_6:36 (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "x")) ($#r1_bcialg_1 :::"<="::: ) (Set (Var "y")))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_bcialg_1 :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y"))))))) ; theorem :: BCIALG_6:37 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) "iff" (Bool (Set ($#k5_bcialg_6 :::"Ker"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) ")" ))) ; theorem :: BCIALG_6:38 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X9")) "," (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ))) "holds" (Bool (Set (Var "g")) "is" ($#v3_funct_2 :::"bijective"::: ) )))) ; theorem :: BCIALG_6:39 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X9")) "," (Set (Var "Y")) "holds" (Bool (Set (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "Y")))))) ; theorem :: BCIALG_6:40 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) (Bool "for" (Set (Var "Z")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X9")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Z"))) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "Z")))))) ; theorem :: BCIALG_6:41 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "holds" (Bool (Set ($#k5_bcialg_6 :::"Ker"::: ) (Set (Var "f"))) "is" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X"))))) ; registrationlet "X", "X9" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "f" be ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Const "X")) "," (Set (Const "X9")); cluster (Set ($#k5_bcialg_6 :::"Ker"::: ) "f") -> ($#v12_bcialg_1 :::"closed"::: ) for ($#m2_bcialg_1 :::"Ideal"::: ) "of" "X"; end; theorem :: BCIALG_6:42 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) )) "holds" (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X9")) (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")))))))) ; theorem :: BCIALG_6:43 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v10_bcialg_1 :::"minimal"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) "is" ($#v10_bcialg_1 :::"minimal"::: ) )))) ; theorem :: BCIALG_6:44 (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_bcialg_1 :::"AtomSet"::: ) (Set (Var "X9"))) "st" (Bool (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "a")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k7_bcialg_1 :::"BranchV"::: ) (Set (Var "b")))))))) ; theorem :: BCIALG_6:45 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "A9")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X9")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "A9")) "is" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X9")))) "holds" (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set (Var "A9"))) "is" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")))))) ; theorem :: BCIALG_6:46 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "A9")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X9")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "A9")) "is" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X9")))) "holds" (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set (Var "A9"))) "is" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")))))) ; theorem :: BCIALG_6:47 (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "I")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "I"))) "is" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X9")))))) ; theorem :: BCIALG_6:48 (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "CI")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "CI"))) "is" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X9")))))) ; definitionlet "X", "X9" be ($#l2_bcialg_1 :::"BCI-algebra":::); pred "X" "," "X9" :::"are_isomorphic"::: means :: BCIALG_6:def 9 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" "X" "," "X9" "st" (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) )); end; :: deftheorem defines :::"are_isomorphic"::: BCIALG_6:def 9 : (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) "holds" (Bool "(" (Bool (Set (Var "X")) "," (Set (Var "X9")) ($#r1_bcialg_6 :::"are_isomorphic"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) )) ")" )); registrationlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "I" be ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RI" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "I")); cluster (Set "X" ($#k9_bcialg_2 :::"./."::: ) "RI") -> ($#v2_bcialg_1 :::"strict"::: ) ($#v3_bcialg_1 :::"being_B"::: ) ($#v4_bcialg_1 :::"being_C"::: ) ($#v5_bcialg_1 :::"being_I"::: ) ($#v7_bcialg_1 :::"being_BCI-4"::: ) ; end; definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "I" be ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RI" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "I")); func :::"nat_hom"::: "RI" -> ($#m1_subset_1 :::"BCI-homomorphism":::) "of" "X" "," (Set "(" "X" ($#k9_bcialg_2 :::"./."::: ) "RI" ")" ) means :: BCIALG_6:def 10 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "X" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_eqrel_1 :::"Class"::: ) "(" "RI" "," (Set (Var "x")) ")" ))); end; :: deftheorem defines :::"nat_hom"::: BCIALG_6:def 10 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "I")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RI")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "I")) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set "(" (Set (Var "X")) ($#k9_bcialg_2 :::"./."::: ) (Set (Var "RI")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k6_bcialg_6 :::"nat_hom"::: ) (Set (Var "RI")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set (Var "RI")) "," (Set (Var "x")) ")" ))) ")" ))))); begin theorem :: BCIALG_6:49 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "I")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RI")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "I")) "holds" (Bool (Set ($#k6_bcialg_6 :::"nat_hom"::: ) (Set (Var "RI"))) "is" ($#v2_funct_2 :::"onto"::: ) )))) ; theorem :: BCIALG_6:50 (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "I")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RI")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "I")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k5_bcialg_6 :::"Ker"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set "(" (Set (Var "X")) ($#k9_bcialg_2 :::"./."::: ) (Set (Var "RI")) ")" ) "," (Set (Var "X9")) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k6_bcialg_6 :::"nat_hom"::: ) (Set (Var "RI")) ")" ))) & (Bool (Set (Var "h")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )))))) ; theorem :: BCIALG_6:51 (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "I")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RI")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "I")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k5_bcialg_6 :::"Ker"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set "(" (Set (Var "X")) ($#k9_bcialg_2 :::"./."::: ) (Set (Var "RI")) ")" ) "," (Set (Var "X9")) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k6_bcialg_6 :::"nat_hom"::: ) (Set (Var "RI")) ")" ))) & (Bool (Set (Var "h")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )))))) ; theorem :: BCIALG_6:52 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) "holds" (Bool (Set ($#k5_bcialg_6 :::"Ker"::: ) (Set "(" ($#k6_bcialg_6 :::"nat_hom"::: ) (Set (Var "RK")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "K")))))) ; begin theorem :: BCIALG_6:53 (Bool "for" (Set (Var "X9")) "," (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "H9")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X9")) (Bool "for" (Set (Var "I")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RI")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "I")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k5_bcialg_6 :::"Ker"::: ) (Set (Var "f")))) & (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "H9"))) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "X")) ($#k9_bcialg_2 :::"./."::: ) (Set (Var "RI"))) "," (Set (Var "H9")) ($#r1_bcialg_6 :::"are_isomorphic"::: ) )))))) ; theorem :: BCIALG_6:54 (Bool "for" (Set (Var "X")) "," (Set (Var "X9")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "I")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RI")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "I")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"BCI-homomorphism":::) "of" (Set (Var "X")) "," (Set (Var "X9")) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k5_bcialg_6 :::"Ker"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) )) "holds" (Bool (Set (Set (Var "X")) ($#k9_bcialg_2 :::"./."::: ) (Set (Var "RI"))) "," (Set (Var "X9")) ($#r1_bcialg_6 :::"are_isomorphic"::: ) ))))) ; begin definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "G" be ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Const "X")); let "K" be ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RK" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "K")); func :::"Union"::: "(" "G" "," "RK" ")" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" "X" equals :: BCIALG_6:def 11 (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k6_eqrel_1 :::"Class"::: ) "(" "RK" "," (Set (Var "a")) ")" ")" ) where a "is" ($#m1_subset_1 :::"Element":::) "of" "G" : (Bool (Set ($#k6_eqrel_1 :::"Class"::: ) "(" "RK" "," (Set (Var "a")) ")" ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" "X" ($#k9_bcialg_2 :::"./."::: ) "RK" ")" ))) "}" ); end; :: deftheorem defines :::"Union"::: BCIALG_6:def 11 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) "holds" (Bool (Set ($#k7_bcialg_6 :::"Union"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k6_eqrel_1 :::"Class"::: ) "(" (Set (Var "RK")) "," (Set (Var "a")) ")" ")" ) where a "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) : (Bool (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set (Var "RK")) "," (Set (Var "a")) ")" ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set (Var "X")) ($#k9_bcialg_2 :::"./."::: ) (Set (Var "RK")) ")" ))) "}" )))))); definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "G" be ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Const "X")); let "K" be ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RK" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "K")); func :::"HKOp"::: "(" "G" "," "RK" ")" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k7_bcialg_6 :::"Union"::: ) "(" "G" "," "RK" ")" ")" ) means :: BCIALG_6:def 12 (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_bcialg_6 :::"Union"::: ) "(" "G" "," "RK" ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "X" "st" (Bool (Bool (Set (Var "w1")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Var "w2")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "holds" (Bool (Set it ($#k5_binop_1 :::"."::: ) "(" (Set (Var "w1")) "," (Set (Var "w2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")))))); end; :: deftheorem defines :::"HKOp"::: BCIALG_6:def 12 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k7_bcialg_6 :::"Union"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k8_bcialg_6 :::"HKOp"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" )) "iff" (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_bcialg_6 :::"Union"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "w1")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Var "w2")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "holds" (Bool (Set (Set (Var "b5")) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "w1")) "," (Set (Var "w2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_bcialg_1 :::"\"::: ) (Set (Var "y")))))) ")" )))))); definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "G" be ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Const "X")); let "K" be ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RK" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "K")); func :::"zeroHK"::: "(" "G" "," "RK" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_bcialg_6 :::"Union"::: ) "(" "G" "," "RK" ")" ) equals :: BCIALG_6:def 13 (Set ($#k4_struct_0 :::"0."::: ) "X"); end; :: deftheorem defines :::"zeroHK"::: BCIALG_6:def 13 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) "holds" (Bool (Set ($#k9_bcialg_6 :::"zeroHK"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X")))))))); definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "G" be ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Const "X")); let "K" be ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RK" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "K")); func :::"HK"::: "(" "G" "," "RK" ")" -> ($#l2_bcialg_1 :::"BCIStr_0"::: ) equals :: BCIALG_6:def 14 (Set ($#g2_bcialg_1 :::"BCIStr_0"::: ) "(#" (Set "(" ($#k7_bcialg_6 :::"Union"::: ) "(" "G" "," "RK" ")" ")" ) "," (Set "(" ($#k8_bcialg_6 :::"HKOp"::: ) "(" "G" "," "RK" ")" ")" ) "," (Set "(" ($#k9_bcialg_6 :::"zeroHK"::: ) "(" "G" "," "RK" ")" ")" ) "#)" ); end; :: deftheorem defines :::"HK"::: BCIALG_6:def 14 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) "holds" (Bool (Set ($#k10_bcialg_6 :::"HK"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_bcialg_1 :::"BCIStr_0"::: ) "(#" (Set "(" ($#k7_bcialg_6 :::"Union"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ")" ) "," (Set "(" ($#k8_bcialg_6 :::"HKOp"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ")" ) "," (Set "(" ($#k9_bcialg_6 :::"zeroHK"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ")" ) "#)" )))))); registrationlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "G" be ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Const "X")); let "K" be ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RK" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "K")); cluster (Set ($#k10_bcialg_6 :::"HK"::: ) "(" "G" "," "RK" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; definitionlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "G" be ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Const "X")); let "K" be ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RK" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "K")); let "w1", "w2" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_bcialg_6 :::"Union"::: ) "(" (Set (Const "G")) "," (Set (Const "RK")) ")" ); func "w1" :::"\"::: "w2" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_bcialg_6 :::"Union"::: ) "(" "G" "," "RK" ")" ) equals :: BCIALG_6:def 15 (Set (Set "(" ($#k8_bcialg_6 :::"HKOp"::: ) "(" "G" "," "RK" ")" ")" ) ($#k5_binop_1 :::"."::: ) "(" "w1" "," "w2" ")" ); end; :: deftheorem defines :::"\"::: BCIALG_6:def 15 : (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_bcialg_6 :::"Union"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) "holds" (Bool (Set (Set (Var "w1")) ($#k11_bcialg_6 :::"\"::: ) (Set (Var "w2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_bcialg_6 :::"HKOp"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "w1")) "," (Set (Var "w2")) ")" ))))))); theorem :: BCIALG_6:55 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) "holds" (Bool (Set ($#k10_bcialg_6 :::"HK"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) "is" ($#l2_bcialg_1 :::"BCI-algebra":::)))))) ; registrationlet "X" be ($#l2_bcialg_1 :::"BCI-algebra":::); let "G" be ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Const "X")); let "K" be ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Const "X")); let "RK" be ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Const "X")) "," (Set (Const "K")); cluster (Set ($#k10_bcialg_6 :::"HK"::: ) "(" "G" "," "RK" ")" ) -> ($#v2_bcialg_1 :::"strict"::: ) ($#v3_bcialg_1 :::"being_B"::: ) ($#v4_bcialg_1 :::"being_C"::: ) ($#v5_bcialg_1 :::"being_I"::: ) ($#v7_bcialg_1 :::"being_BCI-4"::: ) ; end; theorem :: BCIALG_6:56 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) "holds" (Bool (Set ($#k10_bcialg_6 :::"HK"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) "is" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X"))))))) ; theorem :: BCIALG_6:57 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "G"))) ($#k9_subset_1 :::"/\"::: ) (Set (Var "K"))) "is" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "G")))))) ; theorem :: BCIALG_6:58 (Bool "for" (Set (Var "X")) "being" ($#l2_bcialg_1 :::"BCI-algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_bcialg_1 :::"SubAlgebra"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "K")) "being" ($#v12_bcialg_1 :::"closed"::: ) ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "RK")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "X")) "," (Set (Var "K")) (Bool "for" (Set (Var "K1")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set ($#k10_bcialg_6 :::"HK"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) (Bool "for" (Set (Var "RK1")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set ($#k10_bcialg_6 :::"HK"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ) "," (Set (Var "K1")) (Bool "for" (Set (Var "I")) "being" ($#m2_bcialg_1 :::"Ideal"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "RI")) "being" ($#m5_bcialg_2 :::"I-congruence"::: ) "of" (Set (Var "G")) "," (Set (Var "I")) "st" (Bool (Bool (Set (Var "RK1")) ($#r1_hidden :::"="::: ) (Set (Var "RK"))) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "G"))) ($#k9_subset_1 :::"/\"::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "G")) ($#k9_bcialg_2 :::"./."::: ) (Set (Var "RI"))) "," (Set (Set "(" ($#k10_bcialg_6 :::"HK"::: ) "(" (Set (Var "G")) "," (Set (Var "RK")) ")" ")" ) ($#k9_bcialg_2 :::"./."::: ) (Set (Var "RK1"))) ($#r1_bcialg_6 :::"are_isomorphic"::: ) ))))))))) ;