:: DESCIP_1  semantic presentation begin  









registrationlet "n" be   ($#m1_hidden :::"Nat":::); let "f" be (Set (Const "n"))  ($#v3_card_1 :::"-element"::: )    ($#m1_hidden :::"FinSequence":::); 
cluster (Set   ($#k3_finseq_5 :::"Rev"::: )  "f") -> "n"  ($#v3_card_1 :::"-element"::: )   ; 
end; definitionlet "D" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  ; let "n" be   ($#m1_hidden :::"Nat":::); let "f" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Const "D"))); :: original: :::"Rev"::: redefine func  :::"Rev"::: "f" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set "n"  ($#k4_finseq_2 :::"-tuples_on"::: )  "D"); end; notationlet "n" be   ($#m1_hidden :::"Nat":::); let "f" be   ($#m1_hidden :::"FinSequence":::); synonym  :::"Op-Left":::  "(" "f" "," "n" ")"  for "f" :::"|"::: "n"; synonym  :::"Op-Right":::  "(" "f" "," "n" ")"  for "f" :::"/^"::: "n"; end; definitionlet "D" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  ; let "n" be   ($#m1_hidden :::"Nat":::); let "f" be    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Const "D")); :: original: :::"Op-Left"::: redefine func  :::"Op-Left":::  "(" "f" "," "n" ")"  ->    ($#m2_finseq_1 :::"FinSequence"::: )   "of" "D"; :: original: :::"Op-Right"::: redefine func  :::"Op-Right":::  "(" "f" "," "n" ")"  ->    ($#m2_finseq_1 :::"FinSequence"::: )   "of" "D"; end; notationlet "D" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  ; let "n" be   ($#m1_hidden :::"Nat":::); let "s" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Const "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Const "D"))); synonym  :::"SP-Left"::: "s" for  :::"Op-Left":::  "(" "n" "," "D" ")" ; synonym  :::"SP-Right"::: "s" for  :::"Op-Right":::  "(" "n" "," "D" ")" ; end; definitionlet "D" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  ; let "n" be   ($#m1_hidden :::"Nat":::); let "s" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Const "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Const "D"))); :: original: :::"Op-Left"::: redefine func  :::"SP-Left"::: "s" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set "n"  ($#k4_finseq_2 :::"-tuples_on"::: )  "D"); end; 
theorem :: DESCIP_1:1 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "m")) "," (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D")))  "st" (Bool (Bool (Set (Var "m"))  ($#r1_xxreal_0 :::"<="::: )  (Set (Var "n")))) "holds"  (Bool (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set (Var "s")) "," (Set (Var "m")) ")" ) "is"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "m"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D"))))))) ; 
 
theorem :: DESCIP_1:2 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "l")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D")))  "st" (Bool (Bool (Set (Var "m"))  ($#r1_xxreal_0 :::"<="::: )  (Set (Var "n"))) & (Bool (Set (Var "l"))  ($#r1_hidden :::"="::: )  (Set (Set (Var "n"))  ($#k6_xcmplx_0 :::"-"::: )  (Set (Var "m"))))) "holds"  (Bool (Set   ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set (Var "s")) "," (Set (Var "m")) ")" ) "is"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "l"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D"))))))) ; 
 definitionlet "D" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  ; let "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); let "s" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Const "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Const "D"))); :: original: :::"Op-Right"::: redefine func  :::"SP-Right"::: "s" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set "n"  ($#k4_finseq_2 :::"-tuples_on"::: )  "D"); end; 
theorem :: DESCIP_1:3 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Var "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D"))) "holds"  (Bool (Set (Set  "("  ($#k4_descip_1 :::"SP-Left"::: )  (Set (Var "s")) ")" )  ($#k9_binarith :::"^"::: )  (Set  "("  ($#k5_descip_1 :::"SP-Right"::: )  (Set (Var "s")) ")" ))  ($#r1_hidden :::"="::: )  (Set (Var "s")))))) ; 
 definitionlet "s" be   ($#m1_hidden :::"FinSequence":::); func  :::"Op-LeftShift"::: "s" ->   ($#m1_hidden :::"FinSequence":::) equals :: DESCIP_1:def 1 (Set (Set  "(" "s"  ($#k1_rfinseq :::"/^"::: )  (Num 1) ")" )  ($#k7_finseq_1 :::"^"::: )  (Set  ($#k9_finseq_1 :::"<*"::: ) (Set  "(" "s"  ($#k1_funct_1 :::"."::: )  (Num 1) ")" ) ($#k9_finseq_1 :::"*>"::: ) )); end; 





:: deftheorem    defines :::"Op-LeftShift"::: DESCIP_1:def 1 :  (Bool  "for" (Set (Var "s")) "being"   ($#m1_hidden :::"FinSequence":::) "holds"  (Bool (Set   ($#k6_descip_1 :::"Op-LeftShift"::: )  (Set (Var "s")))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set (Var "s"))  ($#k1_rfinseq :::"/^"::: )  (Num 1) ")" )  ($#k7_finseq_1 :::"^"::: )  (Set  ($#k9_finseq_1 :::"<*"::: ) (Set  "(" (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 1) ")" ) ($#k9_finseq_1 :::"*>"::: ) )))); 
 
theorem :: DESCIP_1:4 (Bool  "for" (Set (Var "s")) "being"   ($#m1_hidden :::"FinSequence":::)  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool (Set   ($#k3_finseq_1 :::"len"::: )  (Set  "("  ($#k6_descip_1 :::"Op-LeftShift"::: )  (Set (Var "s")) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) ; 
 
theorem :: DESCIP_1:5 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool  "("  (Bool (Set   ($#k6_descip_1 :::"Op-LeftShift"::: )  (Set (Var "s"))) "is"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))) & (Bool (Set   ($#k3_finseq_1 :::"len"::: )  (Set  "("  ($#k6_descip_1 :::"Op-LeftShift"::: )  (Set (Var "s")) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))  ")" ))) ; 
 
theorem :: DESCIP_1:6 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D"))) "holds"  (Bool (Set   ($#k6_descip_1 :::"Op-LeftShift"::: )  (Set (Var "s"))) "is"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D"))))))) ; 
 definitionlet "s" be   ($#m1_hidden :::"FinSequence":::); func  :::"Op-RightShift"::: "s" ->   ($#m1_hidden :::"FinSequence":::) equals :: DESCIP_1:def 2 (Set (Set  "(" (Set  ($#k9_finseq_1 :::"<*"::: ) (Set  "(" "s"  ($#k1_funct_1 :::"."::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  "s" ")" ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )  ($#k7_finseq_1 :::"^"::: )  "s" ")" )  ($#k16_finseq_1 :::"|"::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  "s" ")" )); end; 





:: deftheorem    defines :::"Op-RightShift"::: DESCIP_1:def 2 :  (Bool  "for" (Set (Var "s")) "being"   ($#m1_hidden :::"FinSequence":::) "holds"  (Bool (Set   ($#k7_descip_1 :::"Op-RightShift"::: )  (Set (Var "s")))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set  ($#k9_finseq_1 :::"<*"::: ) (Set  "(" (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  (Set (Var "s")) ")" ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )  ($#k7_finseq_1 :::"^"::: )  (Set (Var "s")) ")" )  ($#k16_finseq_1 :::"|"::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  (Set (Var "s")) ")" )))); 
 
theorem :: DESCIP_1:7 (Bool  "for" (Set (Var "s")) "being"   ($#m1_hidden :::"FinSequence":::) "holds"  (Bool (Set   ($#k3_finseq_1 :::"len"::: )  (Set  "("  ($#k7_descip_1 :::"Op-RightShift"::: )  (Set (Var "s")) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) ; 
 
theorem :: DESCIP_1:8 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool  "("  (Bool (Set   ($#k7_descip_1 :::"Op-RightShift"::: )  (Set (Var "s"))) "is"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))) & (Bool (Set   ($#k3_finseq_1 :::"len"::: )  (Set  "("  ($#k7_descip_1 :::"Op-RightShift"::: )  (Set (Var "s")) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))  ")" ))) ; 
 
theorem :: DESCIP_1:9 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D"))) "holds"  (Bool (Set   ($#k7_descip_1 :::"Op-RightShift"::: )  (Set (Var "s"))) "is"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D"))))))) ; 
 definitionlet "D" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  ; let "s" be    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Const "D")); let "n" be   ($#m1_hidden :::"Integer":::); assume 
(Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Const "s"))))
 ; func  :::"Op-Shift":::  "(" "s" "," "n" ")"  ->    ($#m2_finseq_1 :::"FinSequence"::: )   "of" "D" means :: DESCIP_1:def 3 (Bool  "("  (Bool (Set   ($#k3_finseq_1 :::"len"::: )  it)  ($#r1_hidden :::"="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  "s")) & (Bool  "("   "for" (Set (Var "i")) "being"   ($#m1_hidden :::"Nat":::)  "st" (Bool (Bool (Set (Var "i"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  "s" ")" )))) "holds"  (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set "s"  ($#k1_funct_1 :::"."::: )  (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set (Var "i"))  ($#k6_xcmplx_0 :::"-"::: )  (Num 1) ")" )  ($#k2_xcmplx_0 :::"+"::: )  "n" ")" )  ($#k6_int_1 :::"mod"::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  "s" ")" ) ")" )  ($#k2_xcmplx_0 :::"+"::: )  (Num 1) ")" )))  ")" )  ")" ); end; 








:: deftheorem    defines :::"Op-Shift"::: DESCIP_1:def 3 :  (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  (Bool  "for" (Set (Var "n")) "being"   ($#m1_hidden :::"Integer":::)  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool  "for" (Set (Var "b4")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D")) "holds"   (Bool  "("  (Bool (Set (Var "b4"))  ($#r1_hidden :::"="::: )  (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Set (Var "n")) ")" )) "iff" (Bool  "("  (Bool (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "b4")))  ($#r1_hidden :::"="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s")))) & (Bool  "("   "for" (Set (Var "i")) "being"   ($#m1_hidden :::"Nat":::)  "st" (Bool (Bool (Set (Var "i"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  (Set (Var "s")) ")" )))) "holds"  (Bool (Set (Set (Var "b4"))  ($#k1_funct_1 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set (Var "i"))  ($#k6_xcmplx_0 :::"-"::: )  (Num 1) ")" )  ($#k2_xcmplx_0 :::"+"::: )  (Set (Var "n")) ")" )  ($#k6_int_1 :::"mod"::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  (Set (Var "s")) ")" ) ")" )  ($#k2_xcmplx_0 :::"+"::: )  (Num 1) ")" )))  ")" )  ")" )  ")" ))))); 
 
theorem :: DESCIP_1:10 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  (Bool  "for" (Set (Var "n")) "," (Set (Var "m")) "being"   ($#m1_hidden :::"Integer":::)  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set  "("  ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Set (Var "n")) ")"  ")" ) "," (Set (Var "m")) ")" )  ($#r1_hidden :::"="::: )  (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Set  "(" (Set (Var "n"))  ($#k2_xcmplx_0 :::"+"::: )  (Set (Var "m")) ")" ) ")" ))))) ; 
 
theorem :: DESCIP_1:11 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Set  ($#k6_numbers :::"0"::: ) ) ")" )  ($#r1_hidden :::"="::: )  (Set (Var "s"))))) ; 
 
theorem :: DESCIP_1:12 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Set  "("  ($#k3_finseq_1 :::"len"::: )  (Set (Var "s")) ")" ) ")" )  ($#r1_hidden :::"="::: )  (Set (Var "s"))))) ; 
 
theorem :: DESCIP_1:13 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Set  "("  ($#k4_xcmplx_0 :::"-"::: )  (Set  "("  ($#k3_finseq_1 :::"len"::: )  (Set (Var "s")) ")" ) ")" ) ")" )  ($#r1_hidden :::"="::: )  (Set (Var "s"))))) ; 
 
theorem :: DESCIP_1:14 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "m")) "being"   ($#m1_hidden :::"Integer":::)  (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D"))) "holds"  (Bool (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Set (Var "m")) ")" ) "is"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Var "D")))))))) ; 
 
theorem :: DESCIP_1:15 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Set  "("  ($#k4_xcmplx_0 :::"-"::: )  (Num 1) ")" ) ")" )  ($#r1_hidden :::"="::: )  (Set   ($#k7_descip_1 :::"Op-RightShift"::: )  (Set (Var "s")))))) ; 
 
theorem :: DESCIP_1:16 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  "st" (Bool (Bool (Num 1)  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set (Var "s")) "," (Num 1) ")" )  ($#r1_hidden :::"="::: )  (Set   ($#k6_descip_1 :::"Op-LeftShift"::: )  (Set (Var "s")))))) ; 
 definitionlet "x", "y" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 28)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); :: original: :::"^"::: redefine func "x" :::"^"::: "y" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 56)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); end; definitionlet "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); let "s" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); let "i" be   ($#m1_hidden :::"Nat":::); :: original: :::"."::: redefine func "s" :::"."::: "i" ->    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k6_margrel1 :::"BOOLEAN"::: )  ); end; definitionlet "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); let "s" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k5_numbers :::"NAT"::: ) )); let "i" be   ($#m1_hidden :::"Nat":::); :: original: :::"."::: redefine func "s" :::"."::: "i" ->    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); end; registrationlet "n" be   ($#m1_hidden :::"Nat":::); 
cluster   ->   ($#v1_margrel1 :::"boolean-valued"::: )    for    ($#m1_subset_1 :::"Element"::: )   "of" (Set "n"  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); 
end; notationlet "n" be    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); let "s", "t" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); synonym  :::"Op-XOR":::  "(" "s" "," "t" ")"  for "n" :::"'xor'"::: "s"; end; definitionlet "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); let "s", "t" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); :: original: :::"Op-XOR"::: redefine func  :::"Op-XOR":::  "(" "s" "," "t" ")"  ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set "n"  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 4 (Bool  "for" (Set (Var "i")) "being"   ($#m1_hidden :::"Nat":::)  "st" (Bool (Bool (Set (Var "i"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  "n"))) "holds"  (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set (Set  "(" "s"  ($#k10_descip_1 :::"."::: )  (Set (Var "i")) ")" )  ($#k2_binarith :::"'xor'"::: )  (Set  "(" "t"  ($#k10_descip_1 :::"."::: )  (Set (Var "i")) ")" )))); commutativity  
(Bool  "for" (Set (Var "b1")) "," (Set (Var "s")) "," (Set (Var "t")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))  "st" (Bool (Bool  "("   "for" (Set (Var "i")) "being"   ($#m1_hidden :::"Nat":::)  "st" (Bool (Bool (Set (Var "i"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Set (Const "n"))))) "holds"  (Bool (Set (Set (Var "b1"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set (Var "s"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")) ")" )  ($#k2_binarith :::"'xor'"::: )  (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")) ")" )))  ")" )) "holds"  (Bool  "for" (Set (Var "i")) "being"   ($#m1_hidden :::"Nat":::)  "st" (Bool (Bool (Set (Var "i"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Set (Const "n"))))) "holds"  (Bool (Set (Set (Var "b1"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")) ")" )  ($#k2_binarith :::"'xor'"::: )  (Set  "(" (Set (Var "s"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")) ")" )))))
 ; end; 







:: deftheorem    defines :::"Op-XOR"::: DESCIP_1:def 4 :  (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "b4")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b4"))  ($#r1_hidden :::"="::: )  (Set   ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set (Var "s")) "," (Set (Var "t")) ")" )) "iff" (Bool  "for" (Set (Var "i")) "being"   ($#m1_hidden :::"Nat":::)  "st" (Bool (Bool (Set (Var "i"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Set (Var "n"))))) "holds"  (Bool (Set (Set (Var "b4"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set (Var "s"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")) ")" )  ($#k2_binarith :::"'xor'"::: )  (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Set (Var "i")) ")" ))))  ")" ))); 
 definitionlet "n", "k" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); let "RK" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Const "k"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )); let "i" be    ($#m2_subset_1 :::"Element"::: )   "of" (Set   ($#k2_finseq_1 :::"Seg"::: )  (Set (Const "k"))); :: original: :::"."::: redefine func "RK" :::"."::: "i" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set "n"  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); end; 
theorem :: DESCIP_1:17 (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "s")) "," (Set (Var "t")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set   ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set (Var "s")) "," (Set (Var "t")) ")"  ")" ) "," (Set (Var "t")) ")" )  ($#r1_hidden :::"="::: )  (Set (Var "s"))))) ; 
 definitionlet "m" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); let "D" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  ; let "L" be   ($#m1_subset_1 :::"sequence":::) "of" (Set  "(" (Set (Const "m"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set (Const "D")) ")" ); let "i" be   ($#m1_hidden :::"Nat":::); :: original: :::"."::: redefine func "L" :::"."::: "i" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set "m"  ($#k4_finseq_2 :::"-tuples_on"::: )  "D"); end; definitionlet "f" be   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16); let "i" be    ($#m1_hidden :::"set"::: )  ; :: original: :::"."::: redefine func "f" :::"."::: "i" ->    ($#m1_subset_1 :::"Element"::: )   "of" (Num 16); end; 
theorem :: DESCIP_1:18 (Bool  "for" (Set (Var "D")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "D"))  (Bool  "for" (Set (Var "n")) "," (Set (Var "m")) "being"   ($#m1_hidden :::"Nat":::)  "st" (Bool (Bool (Set (Set (Var "n"))  ($#k2_xcmplx_0 :::"+"::: )  (Set (Var "m")))  ($#r1_xxreal_0 :::"<="::: )  (Set   ($#k3_finseq_1 :::"len"::: )  (Set (Var "s"))))) "holds"  (Bool (Set (Set  "(" (Set (Var "s"))  ($#k17_finseq_1 :::"|"::: )  (Set (Var "n")) ")" )  ($#k8_finseq_1 :::"^"::: )  (Set  "(" (Set  "(" (Set (Var "s"))  ($#k2_rfinseq :::"/^"::: )  (Set (Var "n")) ")" )  ($#k17_finseq_1 :::"|"::: )  (Set (Var "m")) ")" ))  ($#r1_hidden :::"="::: )  (Set (Set (Var "s"))  ($#k17_finseq_1 :::"|"::: )  (Set  "(" (Set (Var "n"))  ($#k2_xcmplx_0 :::"+"::: )  (Set (Var "m")) ")" )))))) ; 
 scheme  :: DESCIP_1:sch 1 QuadChoiceRec{ F1() ->  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  , F2() ->  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  , F3() ->  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  , F4() ->  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )  , F5() ->    ($#m1_subset_1 :::"Element"::: )   "of" (Set F1 "("  ")" ), F6() ->    ($#m1_subset_1 :::"Element"::: )   "of" (Set F2 "("  ")" ), F7() ->    ($#m1_subset_1 :::"Element"::: )   "of" (Set F3 "("  ")" ), F8() ->    ($#m1_subset_1 :::"Element"::: )   "of" (Set F4 "("  ")" ), P1[   ($#m1_hidden :::"set"::: )   ","    ($#m1_hidden :::"set"::: )   ","    ($#m1_hidden :::"set"::: )   ","    ($#m1_hidden :::"set"::: )   ","    ($#m1_hidden :::"set"::: )   ","    ($#m1_hidden :::"set"::: )   ","    ($#m1_hidden :::"set"::: )   ","    ($#m1_hidden :::"set"::: )   ","    ($#m1_hidden :::"set"::: )  ] } : 
(Bool  "ex" (Set (Var "f")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k5_numbers :::"NAT"::: ) ) "," (Set F1 "("  ")" )(Bool  "ex" (Set (Var "g")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k5_numbers :::"NAT"::: ) ) "," (Set F2 "("  ")" )(Bool  "ex" (Set (Var "h")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k5_numbers :::"NAT"::: ) ) "," (Set F3 "("  ")" )(Bool  "ex" (Set (Var "i")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k5_numbers :::"NAT"::: ) ) "," (Set F4 "("  ")" ) "st"  (Bool  "("  (Bool (Set (Set (Var "f"))  ($#k8_nat_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set F5 "("  ")" )) & (Bool (Set (Set (Var "g"))  ($#k8_nat_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set F6 "("  ")" )) & (Bool (Set (Set (Var "h"))  ($#k8_nat_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set F7 "("  ")" )) & (Bool (Set (Set (Var "i"))  ($#k8_nat_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set F8 "("  ")" )) & (Bool  "("   "for" (Set (Var "n")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ) "holds"  (Bool P1[(Set (Var "n")) "," (Set (Set (Var "f"))  ($#k8_nat_1 :::"."::: )  (Set (Var "n"))) "," (Set (Set (Var "g"))  ($#k8_nat_1 :::"."::: )  (Set (Var "n"))) "," (Set (Set (Var "h"))  ($#k8_nat_1 :::"."::: )  (Set (Var "n"))) "," (Set (Set (Var "i"))  ($#k8_nat_1 :::"."::: )  (Set (Var "n"))) "," (Set (Set (Var "f"))  ($#k8_nat_1 :::"."::: )  (Set  "(" (Set (Var "n"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" )) "," (Set (Set (Var "g"))  ($#k8_nat_1 :::"."::: )  (Set  "(" (Set (Var "n"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" )) "," (Set (Set (Var "h"))  ($#k8_nat_1 :::"."::: )  (Set  "(" (Set (Var "n"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" )) "," (Set (Set (Var "i"))  ($#k8_nat_1 :::"."::: )  (Set  "(" (Set (Var "n"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))])  ")" )  ")" )))))
 provided
(Bool  "for" (Set (Var "n")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "x")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set F1 "("  ")" )  (Bool  "for" (Set (Var "y")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set F2 "("  ")" )  (Bool  "for" (Set (Var "z")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set F3 "("  ")" )  (Bool  "for" (Set (Var "w")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set F4 "("  ")" ) (Bool  "ex" (Set (Var "x1")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set F1 "("  ")" )(Bool  "ex" (Set (Var "y1")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set F2 "("  ")" )(Bool  "ex" (Set (Var "z1")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set F3 "("  ")" )(Bool  "ex" (Set (Var "w1")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set F4 "("  ")" ) "st" (Bool P1[(Set (Var "n")) "," (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set (Var "w")) "," (Set (Var "x1")) "," (Set (Var "y1")) "," (Set (Var "z1")) "," (Set (Var "w1"))]))))))))))
 proof 


end; 
theorem :: DESCIP_1:19 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    "holds"  (Bool  "("   "not" (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Num 16))) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 1)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 2)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 3)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 4)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 5)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 6)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 7)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 8)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 9)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 10)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 11)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 12)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 13)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 14)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 15)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 16))  ")" )) ; 
 
theorem :: DESCIP_1:20 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    "holds"  (Bool  "("   "not" (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Num 32))) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 1)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 2)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 3)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 4)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 5)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 6)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 7)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 8)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 9)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 10)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 11)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 12)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 13)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 14)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 15)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 16)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 17)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 18)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 19)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 20)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 21)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 22)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 23)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 24)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 25)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 26)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 27)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 28)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 29)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 30)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 31)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 32))  ")" )) ; 
 
theorem :: DESCIP_1:21 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    "holds"  (Bool  "("   "not" (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Num 48))) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 1)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 2)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 3)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 4)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 5)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 6)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 7)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 8)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 9)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 10)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 11)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 12)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 13)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 14)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 15)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 16)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 17)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 18)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 19)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 20)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 21)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 22)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 23)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 24)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 25)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 26)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 27)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 28)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 29)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 30)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 31)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 32)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 33)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 34)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 35)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 36)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 37)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 38)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 39)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 40)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 41)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 42)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 43)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 44)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 45)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 46)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 47)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 48))  ")" )) ; 
 
theorem :: DESCIP_1:22 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    "holds"  (Bool  "("   "not" (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Num 56))) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 1)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 2)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 3)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 4)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 5)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 6)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 7)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 8)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 9)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 10)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 11)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 12)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 13)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 14)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 15)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 16)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 17)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 18)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 19)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 20)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 21)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 22)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 23)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 24)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 25)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 26)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 27)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 28)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 29)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 30)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 31)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 32)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 33)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 34)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 35)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 36)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 37)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 38)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 39)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 40)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 41)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 42)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 43)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 44)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 45)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 46)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 47)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 48)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 49)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 50)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 51)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 52)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 53)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 54)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 55)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 56))  ")" )) ; 
 
theorem :: DESCIP_1:23 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    "holds"  (Bool  "("   "not" (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Num 64))) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 1)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 2)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 3)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 4)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 5)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 6)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 7)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 8)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 9)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 10)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 11)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 12)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 13)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 14)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 15)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 16)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 17)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 18)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 19)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 20)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 21)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 22)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 23)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 24)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 25)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 26)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 27)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 28)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 29)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 30)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 31)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 32)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 33)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 34)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 35)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 36)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 37)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 38)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 39)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 40)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 41)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 42)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 43)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 44)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 45)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 46)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 47)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 48)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 49)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 50)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 51)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 52)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 53)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 54)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 55)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 56)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 57)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 58)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 59)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 60)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 61)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 62)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 63)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 64))  ")" )) ; 
 
theorem :: DESCIP_1:24 (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::) "holds"  (Bool (Set (Var "n"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k6_domain_1 :::"{"::: ) (Set  ($#k6_numbers :::"0"::: ) ) ($#k6_domain_1 :::"}"::: ) )  ($#k4_subset_1 :::"\/"::: )  (Set  "(" (Set  "("  ($#k2_finseq_1 :::"Seg"::: )  (Set (Var "n")) ")" )  ($#k7_subset_1 :::"\"::: )  (Set  ($#k1_tarski :::"{"::: ) (Set (Var "n")) ($#k1_tarski :::"}"::: ) ) ")" )))) ; 
 
theorem :: DESCIP_1:25 (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::)  (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    "holds"  (Bool  "("   "not" (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set (Var "n"))) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) "or" (Bool  "("  (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set   ($#k2_finseq_1 :::"Seg"::: )  (Set (Var "n")))) & (Bool (Set (Var "x"))  ($#r1_hidden :::"<>"::: )  (Set (Var "n")))  ")" )  ")" ))) ; 
 
theorem :: DESCIP_1:26 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    "holds"  (Bool  "("   "not" (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Num 16)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 1)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 2)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 3)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 4)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 5)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 6)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 7)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 8)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 9)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 10)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 11)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 12)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 13)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 14)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 15))  ")" )) ; 
 
theorem :: DESCIP_1:27 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    "holds"  (Bool  "("   "not" (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Num 64)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 1)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 2)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 3)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 4)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 5)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 6)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 7)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 8)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 9)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 10)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 11)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 12)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 13)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 14)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 15)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 16)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 17)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 18)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 19)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 20)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 21)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 22)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 23)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 24)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 25)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 26)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 27)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 28)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 29)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 30)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 31)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 32)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 33)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 34)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 35)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 36)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 37)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 38)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 39)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 40)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 41)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 42)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 43)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 44)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 45)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 46)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 47)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 48)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 49)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 50)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 51)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 52)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 53)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 54)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 55)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 56)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 57)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 58)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 59)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 60)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 61)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 62)) "or" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Num 63))  ")" )) ; 
 
theorem :: DESCIP_1:28 (Bool  "for" (Set (Var "S")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Var "S")) (Bool  "ex" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "S")) "st"  (Bool  "("  (Bool (Set (Var "s")) "is" (Num 8)  ($#v3_card_1 :::"-element"::: )  ) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Var "x1"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Var "x2"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Var "x3"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Var "x4"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Var "x5"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Var "x6"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Var "x7"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Var "x8")))  ")" )))) ; 
 
theorem :: DESCIP_1:29 (Bool  "for" (Set (Var "S")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "," (Set (Var "x9")) "," (Set (Var "x10")) "," (Set (Var "x11")) "," (Set (Var "x12")) "," (Set (Var "x13")) "," (Set (Var "x14")) "," (Set (Var "x15")) "," (Set (Var "x16")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Var "S")) (Bool  "ex" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "S")) "st"  (Bool  "("  (Bool (Set (Var "s")) "is" (Num 16)  ($#v3_card_1 :::"-element"::: )  ) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Var "x1"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Var "x2"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Var "x3"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Var "x4"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Var "x5"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Var "x6"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Var "x7"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Var "x8"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Var "x9"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Var "x10"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Var "x11"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Var "x12"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Var "x13"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Var "x14"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Var "x15"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Var "x16")))  ")" )))) ; 
 
theorem :: DESCIP_1:30 (Bool  "for" (Set (Var "S")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "," (Set (Var "x9")) "," (Set (Var "x10")) "," (Set (Var "x11")) "," (Set (Var "x12")) "," (Set (Var "x13")) "," (Set (Var "x14")) "," (Set (Var "x15")) "," (Set (Var "x16")) "," (Set (Var "x17")) "," (Set (Var "x18")) "," (Set (Var "x19")) "," (Set (Var "x20")) "," (Set (Var "x21")) "," (Set (Var "x22")) "," (Set (Var "x23")) "," (Set (Var "x24")) "," (Set (Var "x25")) "," (Set (Var "x26")) "," (Set (Var "x27")) "," (Set (Var "x28")) "," (Set (Var "x29")) "," (Set (Var "x30")) "," (Set (Var "x31")) "," (Set (Var "x32")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Var "S")) (Bool  "ex" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "S")) "st"  (Bool  "("  (Bool (Set (Var "s")) "is" (Num 32)  ($#v3_card_1 :::"-element"::: )  ) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Var "x1"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Var "x2"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Var "x3"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Var "x4"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Var "x5"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Var "x6"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Var "x7"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Var "x8"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Var "x9"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Var "x10"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Var "x11"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Var "x12"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Var "x13"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Var "x14"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Var "x15"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Var "x16"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Var "x17"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Var "x18"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Var "x19"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Var "x20"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Var "x21"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Var "x22"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Var "x23"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Var "x24"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Var "x25"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Var "x26"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Var "x27"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Var "x28"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Var "x29"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Var "x30"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Var "x31"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Var "x32")))  ")" )))) ; 
 
theorem :: DESCIP_1:31 (Bool  "for" (Set (Var "S")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "," (Set (Var "x9")) "," (Set (Var "x10")) "," (Set (Var "x11")) "," (Set (Var "x12")) "," (Set (Var "x13")) "," (Set (Var "x14")) "," (Set (Var "x15")) "," (Set (Var "x16")) "," (Set (Var "x17")) "," (Set (Var "x18")) "," (Set (Var "x19")) "," (Set (Var "x20")) "," (Set (Var "x21")) "," (Set (Var "x22")) "," (Set (Var "x23")) "," (Set (Var "x24")) "," (Set (Var "x25")) "," (Set (Var "x26")) "," (Set (Var "x27")) "," (Set (Var "x28")) "," (Set (Var "x29")) "," (Set (Var "x30")) "," (Set (Var "x31")) "," (Set (Var "x32")) "," (Set (Var "x33")) "," (Set (Var "x34")) "," (Set (Var "x35")) "," (Set (Var "x36")) "," (Set (Var "x37")) "," (Set (Var "x38")) "," (Set (Var "x39")) "," (Set (Var "x40")) "," (Set (Var "x41")) "," (Set (Var "x42")) "," (Set (Var "x43")) "," (Set (Var "x44")) "," (Set (Var "x45")) "," (Set (Var "x46")) "," (Set (Var "x47")) "," (Set (Var "x48")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Var "S")) (Bool  "ex" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "S")) "st"  (Bool  "("  (Bool (Set (Var "s")) "is" (Num 48)  ($#v3_card_1 :::"-element"::: )  ) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Var "x1"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Var "x2"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Var "x3"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Var "x4"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Var "x5"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Var "x6"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Var "x7"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Var "x8"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Var "x9"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Var "x10"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Var "x11"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Var "x12"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Var "x13"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Var "x14"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Var "x15"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Var "x16"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Var "x17"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Var "x18"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Var "x19"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Var "x20"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Var "x21"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Var "x22"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Var "x23"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Var "x24"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Var "x25"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Var "x26"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Var "x27"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Var "x28"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Var "x29"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Var "x30"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Var "x31"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Var "x32"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Var "x33"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Var "x34"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Var "x35"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Var "x36"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Var "x37"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Var "x38"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Var "x39"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Var "x40"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Var "x41"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Var "x42"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Var "x43"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Var "x44"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Var "x45"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Var "x46"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Var "x47"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Var "x48")))  ")" )))) ; 
 
theorem :: DESCIP_1:32 (Bool  "for" (Set (Var "S")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "," (Set (Var "x9")) "," (Set (Var "x10")) "," (Set (Var "x11")) "," (Set (Var "x12")) "," (Set (Var "x13")) "," (Set (Var "x14")) "," (Set (Var "x15")) "," (Set (Var "x16")) "," (Set (Var "x17")) "," (Set (Var "x18")) "," (Set (Var "x19")) "," (Set (Var "x20")) "," (Set (Var "x21")) "," (Set (Var "x22")) "," (Set (Var "x23")) "," (Set (Var "x24")) "," (Set (Var "x25")) "," (Set (Var "x26")) "," (Set (Var "x27")) "," (Set (Var "x28")) "," (Set (Var "x29")) "," (Set (Var "x30")) "," (Set (Var "x31")) "," (Set (Var "x32")) "," (Set (Var "x33")) "," (Set (Var "x34")) "," (Set (Var "x35")) "," (Set (Var "x36")) "," (Set (Var "x37")) "," (Set (Var "x38")) "," (Set (Var "x39")) "," (Set (Var "x40")) "," (Set (Var "x41")) "," (Set (Var "x42")) "," (Set (Var "x43")) "," (Set (Var "x44")) "," (Set (Var "x45")) "," (Set (Var "x46")) "," (Set (Var "x47")) "," (Set (Var "x48")) "," (Set (Var "x49")) "," (Set (Var "x50")) "," (Set (Var "x51")) "," (Set (Var "x52")) "," (Set (Var "x53")) "," (Set (Var "x54")) "," (Set (Var "x55")) "," (Set (Var "x56")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Var "S")) (Bool  "ex" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "S")) "st"  (Bool  "("  (Bool (Set (Var "s")) "is" (Num 56)  ($#v3_card_1 :::"-element"::: )  ) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Var "x1"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Var "x2"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Var "x3"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Var "x4"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Var "x5"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Var "x6"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Var "x7"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Var "x8"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Var "x9"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Var "x10"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Var "x11"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Var "x12"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Var "x13"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Var "x14"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Var "x15"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Var "x16"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Var "x17"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Var "x18"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Var "x19"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Var "x20"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Var "x21"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Var "x22"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Var "x23"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Var "x24"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Var "x25"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Var "x26"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Var "x27"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Var "x28"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Var "x29"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Var "x30"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Var "x31"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Var "x32"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Var "x33"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Var "x34"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Var "x35"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Var "x36"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Var "x37"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Var "x38"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Var "x39"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Var "x40"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Var "x41"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Var "x42"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Var "x43"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Var "x44"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Var "x45"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Var "x46"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Var "x47"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Var "x48"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set (Var "x49"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set (Var "x50"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set (Var "x51"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set (Var "x52"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set (Var "x53"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set (Var "x54"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set (Var "x55"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set (Var "x56")))  ")" )))) ; 
 
theorem :: DESCIP_1:33 (Bool  "for" (Set (Var "S")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "," (Set (Var "x9")) "," (Set (Var "x10")) "," (Set (Var "x11")) "," (Set (Var "x12")) "," (Set (Var "x13")) "," (Set (Var "x14")) "," (Set (Var "x15")) "," (Set (Var "x16")) "," (Set (Var "x17")) "," (Set (Var "x18")) "," (Set (Var "x19")) "," (Set (Var "x20")) "," (Set (Var "x21")) "," (Set (Var "x22")) "," (Set (Var "x23")) "," (Set (Var "x24")) "," (Set (Var "x25")) "," (Set (Var "x26")) "," (Set (Var "x27")) "," (Set (Var "x28")) "," (Set (Var "x29")) "," (Set (Var "x30")) "," (Set (Var "x31")) "," (Set (Var "x32")) "," (Set (Var "x33")) "," (Set (Var "x34")) "," (Set (Var "x35")) "," (Set (Var "x36")) "," (Set (Var "x37")) "," (Set (Var "x38")) "," (Set (Var "x39")) "," (Set (Var "x40")) "," (Set (Var "x41")) "," (Set (Var "x42")) "," (Set (Var "x43")) "," (Set (Var "x44")) "," (Set (Var "x45")) "," (Set (Var "x46")) "," (Set (Var "x47")) "," (Set (Var "x48")) "," (Set (Var "x49")) "," (Set (Var "x50")) "," (Set (Var "x51")) "," (Set (Var "x52")) "," (Set (Var "x53")) "," (Set (Var "x54")) "," (Set (Var "x55")) "," (Set (Var "x56")) "," (Set (Var "x57")) "," (Set (Var "x58")) "," (Set (Var "x59")) "," (Set (Var "x60")) "," (Set (Var "x61")) "," (Set (Var "x62")) "," (Set (Var "x63")) "," (Set (Var "x64")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Var "S")) (Bool  "ex" (Set (Var "s")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set (Var "S")) "st"  (Bool  "("  (Bool (Set (Var "s")) "is" (Num 64)  ($#v3_card_1 :::"-element"::: )  ) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Var "x1"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Var "x2"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Var "x3"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Var "x4"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Var "x5"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Var "x6"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Var "x7"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Var "x8"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Var "x9"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Var "x10"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Var "x11"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Var "x12"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Var "x13"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Var "x14"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Var "x15"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Var "x16"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Var "x17"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Var "x18"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Var "x19"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Var "x20"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Var "x21"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Var "x22"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Var "x23"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Var "x24"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Var "x25"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Var "x26"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Var "x27"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Var "x28"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Var "x29"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Var "x30"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Var "x31"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Var "x32"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Var "x33"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Var "x34"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Var "x35"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Var "x36"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Var "x37"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Var "x38"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Var "x39"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Var "x40"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Var "x41"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Var "x42"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Var "x43"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Var "x44"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Var "x45"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Var "x46"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Var "x47"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Var "x48"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set (Var "x49"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set (Var "x50"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set (Var "x51"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set (Var "x52"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set (Var "x53"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set (Var "x54"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set (Var "x55"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set (Var "x56"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Set (Var "x57"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set (Var "x58"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Set (Var "x59"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Set (Var "x60"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set (Var "x61"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Set (Var "x62"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Set (Var "x63"))) & (Bool (Set (Set (Var "s"))  ($#k1_funct_1 :::"."::: )  (Num 64))  ($#r1_hidden :::"="::: )  (Set (Var "x64")))  ")" )))) ; 
 notationlet "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::); let "i" be    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Const "n")); synonym  :::"ntoSeg"::: "i" for  :::"succ"::: "n"; end; definitionlet "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::); let "i" be    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Const "n")); :: original: :::"ntoSeg"::: redefine func  :::"ntoSeg"::: "i" ->    ($#m2_subset_1 :::"Element"::: )   "of" (Set   ($#k2_finseq_1 :::"Seg"::: )  "n"); end; definitionlet "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::); let "f" be   ($#m1_subset_1 :::"Function":::) "of" (Set (Const "n")) "," (Set  "("  ($#k2_finseq_1 :::"Seg"::: )  (Set (Const "n")) ")" ); attr "f" is  :::"NtoSEG":::  means :: DESCIP_1:def 5 (Bool  "for" (Set (Var "i")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" "n" "holds"  (Bool (Set "f"  ($#k3_funct_2 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set   ($#k16_descip_1 :::"ntoSeg"::: )  (Set (Var "i"))))); end; 





:: deftheorem    defines :::"NtoSEG"::: DESCIP_1:def 5 :  (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::)  (Bool  "for" (Set (Var "f")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set  "("  ($#k2_finseq_1 :::"Seg"::: )  (Set (Var "n")) ")" ) "holds"   (Bool  "("  (Bool (Set (Var "f")) "is"   ($#v1_descip_1 :::"NtoSEG"::: )  ) "iff" (Bool  "for" (Set (Var "i")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Var "n")) "holds"  (Bool (Set (Set (Var "f"))  ($#k3_funct_2 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set   ($#k16_descip_1 :::"ntoSeg"::: )  (Set (Var "i")))))  ")" ))); 
 registrationlet "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::); 
cluster bbbadV1_RELAT_1() bbbadV4_RELAT_1("n") bbbadV5_RELAT_1((Set   ($#k2_finseq_1 :::"Seg"::: )  "n"))   ($#v1_funct_1 :::"Function-like"::: )    ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )  bbbadV1_FINSET_1()   ($#v1_partfun1 :::"total"::: )     ($#v1_funct_2 :::"quasi_total"::: )   bbbadV1_VALUED_0() bbbadV2_VALUED_0() bbbadV3_VALUED_0() bbbadV4_VALUED_0()   ($#v1_descip_1 :::"NtoSEG"::: )    for    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k1_zfmisc_1 :::"bool"::: )  (Set  ($#k2_zfmisc_1 :::"[:"::: ) "n" "," (Set  "("  ($#k2_finseq_1 :::"Seg"::: )  "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); 
end; registrationlet "n" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::); 
cluster   ($#v1_funct_1 :::"Function-like"::: )     ($#v1_funct_2 :::"quasi_total"::: )     ($#v1_descip_1 :::"NtoSEG"::: )    ->   ($#v3_funct_2 :::"bijective"::: )     ($#v1_descip_1 :::"NtoSEG"::: )    for    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k1_zfmisc_1 :::"bool"::: )  (Set  ($#k2_zfmisc_1 :::"[:"::: ) "n" "," (Set  "("  ($#k2_finseq_1 :::"Seg"::: )  "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); 
end; 
theorem :: DESCIP_1:34 (Bool  "for" (Set (Var "n")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )   ($#m1_hidden :::"Nat":::)  (Bool  "for" (Set (Var "f")) "being"   ($#v1_descip_1 :::"NtoSEG"::: )    ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set  "("  ($#k2_finseq_1 :::"Seg"::: )  (Set (Var "n")) ")" )  (Bool  "for" (Set (Var "i")) "being"   ($#m1_hidden :::"Nat":::)  "st" (Bool (Bool (Set (Var "i"))  ($#r1_xxreal_0 :::"<"::: )  (Set (Var "n")))) "holds"  (Bool  "("  (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set (Set (Var "i"))  ($#k1_nat_1 :::"+"::: )  (Num 1))) & (Bool (Set (Var "i"))  ($#r2_hidden :::"in"::: )  (Set   ($#k1_relset_1 :::"dom"::: )  (Set (Var "f"))))  ")" )))) ; 
 
theorem :: DESCIP_1:35 (Bool  "for" (Set (Var "S")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "," (Set (Var "x9")) "," (Set (Var "x10")) "," (Set (Var "x11")) "," (Set (Var "x12")) "," (Set (Var "x13")) "," (Set (Var "x14")) "," (Set (Var "x15")) "," (Set (Var "x16")) "," (Set (Var "x17")) "," (Set (Var "x18")) "," (Set (Var "x19")) "," (Set (Var "x20")) "," (Set (Var "x21")) "," (Set (Var "x22")) "," (Set (Var "x23")) "," (Set (Var "x24")) "," (Set (Var "x25")) "," (Set (Var "x26")) "," (Set (Var "x27")) "," (Set (Var "x28")) "," (Set (Var "x29")) "," (Set (Var "x30")) "," (Set (Var "x31")) "," (Set (Var "x32")) "," (Set (Var "x33")) "," (Set (Var "x34")) "," (Set (Var "x35")) "," (Set (Var "x36")) "," (Set (Var "x37")) "," (Set (Var "x38")) "," (Set (Var "x39")) "," (Set (Var "x40")) "," (Set (Var "x41")) "," (Set (Var "x42")) "," (Set (Var "x43")) "," (Set (Var "x44")) "," (Set (Var "x45")) "," (Set (Var "x46")) "," (Set (Var "x47")) "," (Set (Var "x48")) "," (Set (Var "x49")) "," (Set (Var "x50")) "," (Set (Var "x51")) "," (Set (Var "x52")) "," (Set (Var "x53")) "," (Set (Var "x54")) "," (Set (Var "x55")) "," (Set (Var "x56")) "," (Set (Var "x57")) "," (Set (Var "x58")) "," (Set (Var "x59")) "," (Set (Var "x60")) "," (Set (Var "x61")) "," (Set (Var "x62")) "," (Set (Var "x63")) "," (Set (Var "x64")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set (Var "S")) (Bool  "ex" (Set (Var "f")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Set (Var "S")) "st"  (Bool  "("  (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set (Var "x1"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Var "x2"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Var "x3"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Var "x4"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Var "x5"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Var "x6"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Var "x7"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Var "x8"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Var "x9"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Var "x10"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Var "x11"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Var "x12"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Var "x13"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Var "x14"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Var "x15"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Var "x16"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Var "x17"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Var "x18"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Var "x19"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Var "x20"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Var "x21"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Var "x22"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Var "x23"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Var "x24"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Var "x25"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Var "x26"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Var "x27"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Var "x28"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Var "x29"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Var "x30"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Var "x31"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Var "x32"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Var "x33"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Var "x34"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Var "x35"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Var "x36"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Var "x37"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Var "x38"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Var "x39"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Var "x40"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Var "x41"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Var "x42"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Var "x43"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Var "x44"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Var "x45"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Var "x46"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Var "x47"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Var "x48"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Var "x49"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set (Var "x50"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set (Var "x51"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set (Var "x52"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set (Var "x53"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set (Var "x54"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set (Var "x55"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set (Var "x56"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set (Var "x57"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Set (Var "x58"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set (Var "x59"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Set (Var "x60"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Set (Var "x61"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set (Var "x62"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Set (Var "x63"))) & (Bool (Set (Set (Var "f"))  ($#k1_funct_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Set (Var "x64")))  ")" )))) ; 
 begin  definitionfunc  :::"DES-SBOX1":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) means :: DESCIP_1:def 6 (Bool  "("  (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 13))  ")" ); end; 




:: deftheorem    defines :::"DES-SBOX1"::: DESCIP_1:def 6 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k17_descip_1 :::"DES-SBOX1"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 13))  ")" )  ")" )); 
 definitionfunc  :::"DES-SBOX2":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) means :: DESCIP_1:def 7 (Bool  "("  (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 9))  ")" ); end; 




:: deftheorem    defines :::"DES-SBOX2"::: DESCIP_1:def 7 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k18_descip_1 :::"DES-SBOX2"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 9))  ")" )  ")" )); 
 definitionfunc  :::"DES-SBOX3":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) means :: DESCIP_1:def 8 (Bool  "("  (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 12))  ")" ); end; 




:: deftheorem    defines :::"DES-SBOX3"::: DESCIP_1:def 8 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k19_descip_1 :::"DES-SBOX3"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 12))  ")" )  ")" )); 
 definitionfunc  :::"DES-SBOX4":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) means :: DESCIP_1:def 9 (Bool  "("  (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 14))  ")" ); end; 




:: deftheorem    defines :::"DES-SBOX4"::: DESCIP_1:def 9 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k20_descip_1 :::"DES-SBOX4"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 14))  ")" )  ")" )); 
 definitionfunc  :::"DES-SBOX5":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) means :: DESCIP_1:def 10 (Bool  "("  (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 3))  ")" ); end; 




:: deftheorem    defines :::"DES-SBOX5"::: DESCIP_1:def 10 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k21_descip_1 :::"DES-SBOX5"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 3))  ")" )  ")" )); 
 definitionfunc  :::"DES-SBOX6":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) means :: DESCIP_1:def 11 (Bool  "("  (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 13))  ")" ); end; 




:: deftheorem    defines :::"DES-SBOX6"::: DESCIP_1:def 11 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k22_descip_1 :::"DES-SBOX6"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 13))  ")" )  ")" )); 
 definitionfunc  :::"DES-SBOX7":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) means :: DESCIP_1:def 12 (Bool  "("  (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 12))  ")" ); end; 




:: deftheorem    defines :::"DES-SBOX7"::: DESCIP_1:def 12 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k23_descip_1 :::"DES-SBOX7"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 12))  ")" )  ")" )); 
 definitionfunc  :::"DES-SBOX8":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) means :: DESCIP_1:def 13 (Bool  "("  (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set it  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 11))  ")" ); end; 




:: deftheorem    defines :::"DES-SBOX8"::: DESCIP_1:def 13 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 64) "," (Num 16) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k24_descip_1 :::"DES-SBOX8"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Num 11)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Num 14)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Num 7)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Num 4)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Num 10)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Num 8)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Num 13)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Num 15)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Num 12)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Num 9)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Set   ($#k6_numbers :::"0"::: )  )) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Num 3)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Num 5)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Num 6)) & (Bool (Set (Set (Var "b1"))  ($#k15_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Num 11))  ")" )  ")" )); 
 begin  definitionlet "r" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-IP"::: "r" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 14 (Bool  "("  (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 58))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 50))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 42))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 34))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 60))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 52))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 44))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 36))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 62))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 54))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 46))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 38))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 64))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 56))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 48))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 40))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 57))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 49))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 41))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 33))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 25))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 59))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 51))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 43))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 35))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 61))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 53))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 45))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 37))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 63))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 55))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 47))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 39))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 64))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 7)))  ")" ); end; 





:: deftheorem    defines :::"DES-IP"::: DESCIP_1:def 14 :  (Bool  "for" (Set (Var "r")) "," (Set (Var "b2")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b2"))  ($#r1_hidden :::"="::: )  (Set   ($#k25_descip_1 :::"DES-IP"::: )  (Set (Var "r")))) "iff" (Bool  "("  (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 58))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 50))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 42))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 34))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 60))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 52))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 44))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 36))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 62))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 54))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 46))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 38))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 64))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 56))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 48))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 40))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 57))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 49))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 41))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 33))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 25))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 59))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 51))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 43))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 35))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 61))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 53))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 45))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 37))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 63))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 55))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 47))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 39))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 64))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 7)))  ")" )  ")" )); 
 definitionfunc  :::"DES-PIP":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) means :: DESCIP_1:def 15 (Bool  "for" (Set (Var "i")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set it  ($#k3_funct_2 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set   ($#k25_descip_1 :::"DES-IP"::: )  (Set (Var "i"))))); end; 




:: deftheorem    defines :::"DES-PIP"::: DESCIP_1:def 15 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k26_descip_1 :::"DES-PIP"::: )  )) "iff" (Bool  "for" (Set (Var "i")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set (Set (Var "b1"))  ($#k3_funct_2 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set   ($#k25_descip_1 :::"DES-IP"::: )  (Set (Var "i")))))  ")" )); 
 definitionlet "r" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-IPINV"::: "r" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 16 (Bool  "("  (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 40))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 48))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 56))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 64))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 39))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 47))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 55))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 63))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 38))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 46))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 54))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 62))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 37))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 45))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 53))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 61))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 36))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 44))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 52))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 60))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 35))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 43))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 51))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 59))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 34))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 42))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 50))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 58))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 33))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 41))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 49))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 57))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 64))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 25)))  ")" ); end; 





:: deftheorem    defines :::"DES-IPINV"::: DESCIP_1:def 16 :  (Bool  "for" (Set (Var "r")) "," (Set (Var "b2")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b2"))  ($#r1_hidden :::"="::: )  (Set   ($#k27_descip_1 :::"DES-IPINV"::: )  (Set (Var "r")))) "iff" (Bool  "("  (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 40))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 48))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 56))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 64))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 39))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 47))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 55))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 63))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 38))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 46))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 54))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 62))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 37))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 45))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 53))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 61))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 36))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 44))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 52))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 60))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 35))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 43))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 51))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 59))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 34))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 42))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 50))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 58))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 57))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 33))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 58))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 59))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 41))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 60))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 61))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 49))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 62))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 63))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 57))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 64))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 25)))  ")" )  ")" )); 
 definitionfunc  :::"DES-PIPINV":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) means :: DESCIP_1:def 17 (Bool  "for" (Set (Var "i")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set it  ($#k3_funct_2 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set   ($#k27_descip_1 :::"DES-IPINV"::: )  (Set (Var "i"))))); end; 




:: deftheorem    defines :::"DES-PIPINV"::: DESCIP_1:def 17 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k28_descip_1 :::"DES-PIPINV"::: )  )) "iff" (Bool  "for" (Set (Var "i")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set (Set (Var "b1"))  ($#k3_funct_2 :::"."::: )  (Set (Var "i")))  ($#r1_hidden :::"="::: )  (Set   ($#k27_descip_1 :::"DES-IPINV"::: )  (Set (Var "i")))))  ")" )); 
 registration
cluster (Set   ($#k26_descip_1 :::"DES-PIP"::: )  ) ->   ($#v3_funct_2 :::"bijective"::: )   ; 
end; registration
cluster (Set   ($#k28_descip_1 :::"DES-PIPINV"::: )  ) ->   ($#v3_funct_2 :::"bijective"::: )   ; 
end; 
theorem :: DESCIP_1:36 (Bool (Set   ($#k28_descip_1 :::"DES-PIPINV"::: )  )  ($#r2_funct_2 :::"="::: )  (Set (Set  ($#k26_descip_1 :::"DES-PIP"::: ) )  ($#k2_funct_2 :::"""::: )  )) ; 
 begin  definitionlet "r" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-E"::: "r" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 18 (Bool  "("  (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 25))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 25))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 1)))  ")" ); end; 





:: deftheorem    defines :::"DES-E"::: DESCIP_1:def 18 :  (Bool  "for" (Set (Var "r")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))  (Bool  "for" (Set (Var "b2")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b2"))  ($#r1_hidden :::"="::: )  (Set   ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "r")))) "iff" (Bool  "("  (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 25))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 25))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 1)))  ")" )  ")" ))); 
 definitionlet "r" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-P"::: "r" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 19 (Bool  "("  (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 25)))  ")" ); end; 





:: deftheorem    defines :::"DES-P"::: DESCIP_1:def 19 :  (Bool  "for" (Set (Var "r")) "," (Set (Var "b2")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b2"))  ($#r1_hidden :::"="::: )  (Set   ($#k30_descip_1 :::"DES-P"::: )  (Set (Var "r")))) "iff" (Bool  "("  (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 32))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 25)))  ")" )  ")" )); 
 definitionlet "r" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-DIV8"::: "r" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 8)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Num 6)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )) means :: DESCIP_1:def 20 (Bool  "("  (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" "r" "," (Num 6) ")" )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" "r" "," (Num 6) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" "r" "," (Num 12) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" "r" "," (Num 18) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" "r" "," (Num 24) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" "r" "," (Num 30) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" "r" "," (Num 36) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set   ($#k3_descip_1 :::"Op-Right"::: )   "(" "r" "," (Num 42) ")" ))  ")" ); end; 





:: deftheorem    defines :::"DES-DIV8"::: DESCIP_1:def 20 :  (Bool  "for" (Set (Var "r")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))  (Bool  "for" (Set (Var "b2")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 8)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Num 6)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )) "holds"   (Bool  "("  (Bool (Set (Var "b2"))  ($#r1_hidden :::"="::: )  (Set   ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")))) "iff" (Bool  "("  (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set (Var "r")) "," (Num 6) ")" )) & (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set (Var "r")) "," (Num 6) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set (Var "r")) "," (Num 12) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set (Var "r")) "," (Num 18) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set (Var "r")) "," (Num 24) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set (Var "r")) "," (Num 30) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set (Var "r")) "," (Num 36) ")"  ")" ) "," (Num 6) ")" )) & (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set   ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set (Var "r")) "," (Num 42) ")" ))  ")" )  ")" ))); 
 
theorem :: DESCIP_1:37 (Bool  "for" (Set (Var "r")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) (Bool  "ex" (Set (Var "s1")) "," (Set (Var "s2")) "," (Set (Var "s3")) "," (Set (Var "s4")) "," (Set (Var "s5")) "," (Set (Var "s6")) "," (Set (Var "s7")) "," (Set (Var "s8")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 6)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "st"  (Bool  "("  (Bool (Set (Var "s1"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")) ")" )  ($#k1_funct_1 :::"."::: )  (Num 1))) & (Bool (Set (Var "s2"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")) ")" )  ($#k1_funct_1 :::"."::: )  (Num 2))) & (Bool (Set (Var "s3"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")) ")" )  ($#k1_funct_1 :::"."::: )  (Num 3))) & (Bool (Set (Var "s4"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")) ")" )  ($#k1_funct_1 :::"."::: )  (Num 4))) & (Bool (Set (Var "s5"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")) ")" )  ($#k1_funct_1 :::"."::: )  (Num 5))) & (Bool (Set (Var "s6"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")) ")" )  ($#k1_funct_1 :::"."::: )  (Num 6))) & (Bool (Set (Var "s7"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")) ")" )  ($#k1_funct_1 :::"."::: )  (Num 7))) & (Bool (Set (Var "s8"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set (Var "r")) ")" )  ($#k1_funct_1 :::"."::: )  (Num 8))) & (Bool (Set (Var "r"))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set (Var "s1"))  ($#k9_binarith :::"^"::: )  (Set (Var "s2")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "s3")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "s4")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "s5")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "s6")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "s7")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "s8"))))  ")" ))) ; 
 definitionlet "t" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 6)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"B6toN64"::: "t" ->    ($#m1_subset_1 :::"Element"::: )   "of" (Num 64) equals :: DESCIP_1:def 21 (Set (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Num 32)  ($#k4_nat_1 :::"*"::: )  (Set  "(" "t"  ($#k10_descip_1 :::"."::: )  (Num 1) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 16)  ($#k4_nat_1 :::"*"::: )  (Set  "(" "t"  ($#k10_descip_1 :::"."::: )  (Num 6) ")" ) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 8)  ($#k4_nat_1 :::"*"::: )  (Set  "(" "t"  ($#k10_descip_1 :::"."::: )  (Num 2) ")" ) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 4)  ($#k4_nat_1 :::"*"::: )  (Set  "(" "t"  ($#k10_descip_1 :::"."::: )  (Num 3) ")" ) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set  "(" "t"  ($#k10_descip_1 :::"."::: )  (Num 4) ")" ) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 1)  ($#k4_nat_1 :::"*"::: )  (Set  "(" "t"  ($#k10_descip_1 :::"."::: )  (Num 5) ")" ) ")" )); end; 





:: deftheorem    defines :::"B6toN64"::: DESCIP_1:def 21 :  (Bool  "for" (Set (Var "t")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 6)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set   ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "t")))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Num 32)  ($#k4_nat_1 :::"*"::: )  (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Num 1) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 16)  ($#k4_nat_1 :::"*"::: )  (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Num 6) ")" ) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 8)  ($#k4_nat_1 :::"*"::: )  (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Num 2) ")" ) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 4)  ($#k4_nat_1 :::"*"::: )  (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Num 3) ")" ) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Num 4) ")" ) ")" ) ")" )  ($#k2_nat_1 :::"+"::: )  (Set  "(" (Num 1)  ($#k4_nat_1 :::"*"::: )  (Set  "(" (Set (Var "t"))  ($#k10_descip_1 :::"."::: )  (Num 5) ")" ) ")" )))); 
 definitionfunc  :::"N16toB4":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Num 16) "," (Set  "(" (Num 4)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) means :: DESCIP_1:def 22 (Bool  "("  (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Num 1) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Num 1) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Num 1) "," (Num 1) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) ))  ")" ); end; 




:: deftheorem    defines :::"N16toB4"::: DESCIP_1:def 22 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Num 16) "," (Set  "(" (Num 4)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k33_descip_1 :::"N16toB4"::: )  )) "iff" (Bool  "("  (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Num 1) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Num 1) "," (Num 1) "," (Set  ($#k6_numbers :::"0"::: ) ) ($#k9_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set  ($#k9_finseq_4 :::"<*"::: ) (Num 1) "," (Num 1) "," (Num 1) "," (Num 1) ($#k9_finseq_4 :::"*>"::: ) ))  ")" )  ")" )); 
 definitionlet "R" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); let "RKey" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-F":::  "(" "R" "," "RKey" ")"  ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 23 (Bool  "ex" (Set (Var "D1")) "," (Set (Var "D2")) "," (Set (Var "D3")) "," (Set (Var "D4")) "," (Set (Var "D5")) "," (Set (Var "D6")) "," (Set (Var "D7")) "," (Set (Var "D8")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 6)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))(Bool  "ex" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 4)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))(Bool  "ex" (Set (Var "C32")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "st"  (Bool  "("  (Bool (Set (Var "D1"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 1))) & (Bool (Set (Var "D2"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 2))) & (Bool (Set (Var "D3"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 3))) & (Bool (Set (Var "D4"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 4))) & (Bool (Set (Var "D5"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 5))) & (Bool (Set (Var "D6"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 6))) & (Bool (Set (Var "D7"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 7))) & (Bool (Set (Var "D8"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 8))) & (Bool (Set   ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  "R" ")" ) "," "RKey" ")" )  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set (Var "D1"))  ($#k9_binarith :::"^"::: )  (Set (Var "D2")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D3")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D4")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D5")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D6")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D7")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D8")))) & (Bool (Set (Var "x1"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k17_descip_1 :::"DES-SBOX1"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D1")) ")" ) ")" ))) & (Bool (Set (Var "x2"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k18_descip_1 :::"DES-SBOX2"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D2")) ")" ) ")" ))) & (Bool (Set (Var "x3"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k19_descip_1 :::"DES-SBOX3"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D3")) ")" ) ")" ))) & (Bool (Set (Var "x4"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k20_descip_1 :::"DES-SBOX4"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D4")) ")" ) ")" ))) & (Bool (Set (Var "x5"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k21_descip_1 :::"DES-SBOX5"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D5")) ")" ) ")" ))) & (Bool (Set (Var "x6"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k22_descip_1 :::"DES-SBOX6"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D6")) ")" ) ")" ))) & (Bool (Set (Var "x7"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k23_descip_1 :::"DES-SBOX7"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D7")) ")" ) ")" ))) & (Bool (Set (Var "x8"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k24_descip_1 :::"DES-SBOX8"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D8")) ")" ) ")" ))) & (Bool (Set (Var "C32"))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set (Var "x1"))  ($#k9_binarith :::"^"::: )  (Set (Var "x2")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x3")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x4")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x5")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x6")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x7")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x8")))) & (Bool it  ($#r1_hidden :::"="::: )  (Set   ($#k30_descip_1 :::"DES-P"::: )  (Set (Var "C32"))))  ")" )))); end; 






:: deftheorem    defines :::"DES-F"::: DESCIP_1:def 23 :  (Bool  "for" (Set (Var "R")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))  (Bool  "for" (Set (Var "RKey")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))  (Bool  "for" (Set (Var "b3")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b3"))  ($#r1_hidden :::"="::: )  (Set   ($#k34_descip_1 :::"DES-F"::: )   "(" (Set (Var "R")) "," (Set (Var "RKey")) ")" )) "iff" (Bool  "ex" (Set (Var "D1")) "," (Set (Var "D2")) "," (Set (Var "D3")) "," (Set (Var "D4")) "," (Set (Var "D5")) "," (Set (Var "D6")) "," (Set (Var "D7")) "," (Set (Var "D8")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 6)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))(Bool  "ex" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 4)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))(Bool  "ex" (Set (Var "C32")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "st"  (Bool  "("  (Bool (Set (Var "D1"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 1))) & (Bool (Set (Var "D2"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 2))) & (Bool (Set (Var "D3"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 3))) & (Bool (Set (Var "D4"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 4))) & (Bool (Set (Var "D5"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 5))) & (Bool (Set (Var "D6"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 6))) & (Bool (Set (Var "D7"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 7))) & (Bool (Set (Var "D8"))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k31_descip_1 :::"DES-DIV8"::: )  (Set  "("  ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")"  ")" ) ")" )  ($#k1_funct_1 :::"."::: )  (Num 8))) & (Bool (Set   ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "("  ($#k29_descip_1 :::"DES-E"::: )  (Set (Var "R")) ")" ) "," (Set (Var "RKey")) ")" )  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set (Var "D1"))  ($#k9_binarith :::"^"::: )  (Set (Var "D2")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D3")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D4")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D5")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D6")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D7")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "D8")))) & (Bool (Set (Var "x1"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k17_descip_1 :::"DES-SBOX1"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D1")) ")" ) ")" ))) & (Bool (Set (Var "x2"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k18_descip_1 :::"DES-SBOX2"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D2")) ")" ) ")" ))) & (Bool (Set (Var "x3"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k19_descip_1 :::"DES-SBOX3"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D3")) ")" ) ")" ))) & (Bool (Set (Var "x4"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k20_descip_1 :::"DES-SBOX4"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D4")) ")" ) ")" ))) & (Bool (Set (Var "x5"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k21_descip_1 :::"DES-SBOX5"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D5")) ")" ) ")" ))) & (Bool (Set (Var "x6"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k22_descip_1 :::"DES-SBOX6"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D6")) ")" ) ")" ))) & (Bool (Set (Var "x7"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k23_descip_1 :::"DES-SBOX7"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D7")) ")" ) ")" ))) & (Bool (Set (Var "x8"))  ($#r1_hidden :::"="::: )  (Set (Set  ($#k33_descip_1 :::"N16toB4"::: ) )  ($#k3_funct_2 :::"."::: )  (Set  "(" (Set  ($#k24_descip_1 :::"DES-SBOX8"::: ) )  ($#k15_descip_1 :::"."::: )  (Set  "("  ($#k32_descip_1 :::"B6toN64"::: )  (Set (Var "D8")) ")" ) ")" ))) & (Bool (Set (Var "C32"))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set  "(" (Set (Var "x1"))  ($#k9_binarith :::"^"::: )  (Set (Var "x2")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x3")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x4")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x5")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x6")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x7")) ")" )  ($#k9_binarith :::"^"::: )  (Set (Var "x8")))) & (Bool (Set (Var "b3"))  ($#r1_hidden :::"="::: )  (Set   ($#k30_descip_1 :::"DES-P"::: )  (Set (Var "C32"))))  ")" ))))  ")" )))); 
 definitionfunc  :::"DES-FFUNC":::  ->   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) means :: DESCIP_1:def 24 (Bool  "for" (Set (Var "z")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "holds"  (Bool (Set it  ($#k3_funct_2 :::"."::: )  (Set (Var "z")))  ($#r1_hidden :::"="::: )  (Set   ($#k34_descip_1 :::"DES-F"::: )   "(" (Set  "(" (Set (Var "z"))  ($#k2_domain_1 :::"`1"::: )  ")" ) "," (Set  "(" (Set (Var "z"))  ($#k3_domain_1 :::"`2"::: )  ")" ) ")" ))); end; 




:: deftheorem    defines :::"DES-FFUNC"::: DESCIP_1:def 24 :  (Bool  "for" (Set (Var "b1")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k35_descip_1 :::"DES-FFUNC"::: )  )) "iff" (Bool  "for" (Set (Var "z")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "holds"  (Bool (Set (Set (Var "b1"))  ($#k3_funct_2 :::"."::: )  (Set (Var "z")))  ($#r1_hidden :::"="::: )  (Set   ($#k34_descip_1 :::"DES-F"::: )   "(" (Set  "(" (Set (Var "z"))  ($#k2_domain_1 :::"`1"::: )  ")" ) "," (Set  "(" (Set (Var "z"))  ($#k3_domain_1 :::"`2"::: )  ")" ) ")" )))  ")" )); 
 begin  definitionlet "r" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-PC1"::: "r" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 56)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 25 (Bool  "("  (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 57))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 49))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 41))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 33))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 25))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 58))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 50))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 42))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 34))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 59))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 51))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 43))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 35))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 60))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 52))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 44))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 36))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 63))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 55))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 47))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 39))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 62))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 54))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 46))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 38))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 61))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 53))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 45))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 37))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 4)))  ")" ); end; 





:: deftheorem    defines :::"DES-PC1"::: DESCIP_1:def 25 :  (Bool  "for" (Set (Var "r")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))  (Bool  "for" (Set (Var "b2")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 56)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b2"))  ($#r1_hidden :::"="::: )  (Set   ($#k36_descip_1 :::"DES-PC1"::: )  (Set (Var "r")))) "iff" (Bool  "("  (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 57))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 49))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 41))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 33))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 25))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 9))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 58))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 50))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 42))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 34))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 18))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 59))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 51))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 43))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 35))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 60))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 52))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 44))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 36))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 63))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 55))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 47))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 39))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 62))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 54))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 46))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 38))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 22))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 61))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 53))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 45))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 37))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 49))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 50))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 51))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 52))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 53))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 54))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 55))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 56))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 4)))  ")" )  ")" ))); 
 definitionlet "r" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 56)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-PC2"::: "r" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 26 (Bool  "("  (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 41))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 52))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 37))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 47))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 55))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 40))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 51))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 45))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 33))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 48))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 44))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 49))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 39))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 56))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 34))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 53))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 46))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 42))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 50))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 36))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set it  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set "r"  ($#k10_descip_1 :::"."::: )  (Num 32)))  ")" ); end; 





:: deftheorem    defines :::"DES-PC2"::: DESCIP_1:def 26 :  (Bool  "for" (Set (Var "r")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 56)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))  (Bool  "for" (Set (Var "b2")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b2"))  ($#r1_hidden :::"="::: )  (Set   ($#k37_descip_1 :::"DES-PC2"::: )  (Set (Var "r")))) "iff" (Bool  "("  (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 14))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 17))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 11))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 24))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 1))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 5))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 3))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 28))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 15))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 6))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 21))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 10))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 23))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 19))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 12))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 4))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 17))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 26))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 18))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 8))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 19))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 16))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 20))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 7))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 21))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 27))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 22))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 20))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 23))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 13))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 24))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 2))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 25))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 41))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 26))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 52))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 27))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 31))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 28))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 37))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 29))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 47))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 30))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 55))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 31))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 30))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 32))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 40))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 33))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 51))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 34))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 45))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 35))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 33))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 36))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 48))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 37))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 44))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 38))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 49))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 39))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 39))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 40))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 56))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 41))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 34))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 42))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 53))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 43))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 46))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 44))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 42))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 45))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 50))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 46))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 36))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 47))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 29))) & (Bool (Set (Set (Var "b2"))  ($#k10_descip_1 :::"."::: )  (Num 48))  ($#r1_hidden :::"="::: )  (Set (Set (Var "r"))  ($#k10_descip_1 :::"."::: )  (Num 32)))  ")" )  ")" ))); 
 definitionfunc  :::"bitshift_DES":::  ->    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ) means :: DESCIP_1:def 27 (Bool  "("  (Bool it "is" (Num 16)  ($#v3_card_1 :::"-element"::: )  ) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 1))  ")" ); end; 




:: deftheorem    defines :::"bitshift_DES"::: DESCIP_1:def 27 :  (Bool  "for" (Set (Var "b1")) "being"    ($#m2_finseq_1 :::"FinSequence"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ) "holds"   (Bool  "("  (Bool (Set (Var "b1"))  ($#r1_hidden :::"="::: )  (Set   ($#k38_descip_1 :::"bitshift_DES"::: )  )) "iff" (Bool  "("  (Bool (Set (Var "b1")) "is" (Num 16)  ($#v3_card_1 :::"-element"::: )  ) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 1))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 2))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 3))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 4))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 5))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 6))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 7))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 8))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 9))  ($#r1_hidden :::"="::: )  (Num 1)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 10))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 11))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 12))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 13))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 14))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 15))  ($#r1_hidden :::"="::: )  (Num 2)) & (Bool (Set (Set (Var "b1"))  ($#k1_funct_1 :::"."::: )  (Num 16))  ($#r1_hidden :::"="::: )  (Num 1))  ")" )  ")" )); 
 definitionlet "Key" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-KS"::: "Key" ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 16)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )) means :: DESCIP_1:def 28 (Bool  "ex" (Set (Var "C")) "," (Set (Var "D")) "being"   ($#m1_subset_1 :::"sequence":::) "of" (Set  "(" (Num 28)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "st"  (Bool  "("  (Bool (Set (Set (Var "C"))  ($#k14_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k36_descip_1 :::"DES-PC1"::: )  "Key" ")" ) "," (Num 28) ")" )) & (Bool (Set (Set (Var "D"))  ($#k14_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set  "("  ($#k36_descip_1 :::"DES-PC1"::: )  "Key" ")" ) "," (Num 28) ")" )) & (Bool  "("   "for" (Set (Var "i")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  "st" (Bool (Bool (Set   ($#k6_numbers :::"0"::: )  )  ($#r1_xxreal_0 :::"<="::: )  (Set (Var "i"))) & (Bool (Set (Var "i"))  ($#r1_xxreal_0 :::"<="::: )  (Num 15))) "holds"  (Bool  "("  (Bool (Set it  ($#k1_funct_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k37_descip_1 :::"DES-PC2"::: )  (Set  "(" (Set  "(" (Set (Var "C"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ) ")" )  ($#k9_descip_1 :::"^"::: )  (Set  "(" (Set (Var "D"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ) ")" ) ")" ))) & (Bool (Set (Set (Var "C"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set  "(" (Set (Var "C"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")) ")" ) "," (Set  "(" (Set  ($#k38_descip_1 :::"bitshift_DES"::: ) )  ($#k1_funct_1 :::"."::: )  (Set (Var "i")) ")" ) ")" )) & (Bool (Set (Set (Var "D"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set  "(" (Set (Var "D"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")) ")" ) "," (Set  "(" (Set  ($#k38_descip_1 :::"bitshift_DES"::: ) )  ($#k1_funct_1 :::"."::: )  (Set (Var "i")) ")" ) ")" ))  ")" )  ")" )  ")" )); end; 





:: deftheorem    defines :::"DES-KS"::: DESCIP_1:def 28 :  (Bool  "for" (Set (Var "Key")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ))  (Bool  "for" (Set (Var "b2")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 16)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )) "holds"   (Bool  "("  (Bool (Set (Var "b2"))  ($#r1_hidden :::"="::: )  (Set   ($#k39_descip_1 :::"DES-KS"::: )  (Set (Var "Key")))) "iff" (Bool  "ex" (Set (Var "C")) "," (Set (Var "D")) "being"   ($#m1_subset_1 :::"sequence":::) "of" (Set  "(" (Num 28)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "st"  (Bool  "("  (Bool (Set (Set (Var "C"))  ($#k14_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k2_descip_1 :::"Op-Left"::: )   "(" (Set  "("  ($#k36_descip_1 :::"DES-PC1"::: )  (Set (Var "Key")) ")" ) "," (Num 28) ")" )) & (Bool (Set (Set (Var "D"))  ($#k14_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k3_descip_1 :::"Op-Right"::: )   "(" (Set  "("  ($#k36_descip_1 :::"DES-PC1"::: )  (Set (Var "Key")) ")" ) "," (Num 28) ")" )) & (Bool  "("   "for" (Set (Var "i")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  "st" (Bool (Bool (Set   ($#k6_numbers :::"0"::: )  )  ($#r1_xxreal_0 :::"<="::: )  (Set (Var "i"))) & (Bool (Set (Var "i"))  ($#r1_xxreal_0 :::"<="::: )  (Num 15))) "holds"  (Bool  "("  (Bool (Set (Set (Var "b2"))  ($#k1_funct_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k37_descip_1 :::"DES-PC2"::: )  (Set  "(" (Set  "(" (Set (Var "C"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ) ")" )  ($#k9_descip_1 :::"^"::: )  (Set  "(" (Set (Var "D"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ) ")" ) ")" ))) & (Bool (Set (Set (Var "C"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set  "(" (Set (Var "C"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")) ")" ) "," (Set  "(" (Set  ($#k38_descip_1 :::"bitshift_DES"::: ) )  ($#k1_funct_1 :::"."::: )  (Set (Var "i")) ")" ) ")" )) & (Bool (Set (Set (Var "D"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k8_descip_1 :::"Op-Shift"::: )   "(" (Set  "(" (Set (Var "D"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")) ")" ) "," (Set  "(" (Set  ($#k38_descip_1 :::"bitshift_DES"::: ) )  ($#k1_funct_1 :::"."::: )  (Set (Var "i")) ")" ) ")" ))  ")" )  ")" )  ")" ))  ")" ))); 
 begin  definitionlet "n", "m", "k" be  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  ); let "RK" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Const "k"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Set (Const "m"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )); let "F" be   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Set (Const "m"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set  "(" (Set (Const "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ); let "IP" be   ($#m1_subset_1 :::"Permutation":::) "of" (Set  "(" (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Const "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ); let "M" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Const "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-like-CoDec":::  "(" "M" "," "F" "," "IP" "," "RK" ")"  ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  "n" ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 29 (Bool  "ex" (Set (Var "L")) "," (Set (Var "R")) "being"   ($#m1_subset_1 :::"sequence":::) "of" (Set  "(" "n"  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "st"  (Bool  "("  (Bool (Set (Set (Var "L"))  ($#k14_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k4_descip_1 :::"SP-Left"::: )  (Set  "(" "IP"  ($#k3_funct_2 :::"."::: )  "M" ")" ))) & (Bool (Set (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k5_descip_1 :::"SP-Right"::: )  (Set  "(" "IP"  ($#k3_funct_2 :::"."::: )  "M" ")" ))) & (Bool  "("   "for" (Set (Var "i")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  "st" (Bool (Bool (Set   ($#k6_numbers :::"0"::: )  )  ($#r1_xxreal_0 :::"<="::: )  (Set (Var "i"))) & (Bool (Set (Var "i"))  ($#r1_xxreal_0 :::"<="::: )  (Set "k"  ($#k6_xcmplx_0 :::"-"::: )  (Num 1)))) "holds"  (Bool  "("  (Bool (Set (Set (Var "L"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")))) & (Bool (Set (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "(" (Set (Var "L"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")) ")" ) "," (Set  "(" "F"  ($#k2_binop_1 :::"."::: )   "(" (Set  "(" (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")) ")" ) "," (Set  "(" "RK"  ($#k7_partfun1 :::"/."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ) ")" ) ")"  ")" ) ")" ))  ")" )  ")" ) & (Bool it  ($#r1_hidden :::"="::: )  (Set (Set  "(" "IP"  ($#k2_funct_2 :::"""::: )  ")" )  ($#k1_funct_1 :::"."::: )  (Set  "(" (Set  "(" (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  "k" ")" )  ($#k9_binarith :::"^"::: )  (Set  "(" (Set (Var "L"))  ($#k14_descip_1 :::"."::: )  "k" ")" ) ")" )))  ")" )); end; 











:: deftheorem    defines :::"DES-like-CoDec"::: DESCIP_1:def 29 :  (Bool  "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "k")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "RK")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "k"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Set (Var "m"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ))  (Bool  "for" (Set (Var "F")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Set (Var "m"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set  "(" (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )  (Bool  "for" (Set (Var "IP")) "being"   ($#m1_subset_1 :::"Permutation":::) "of" (Set  "(" (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Var "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )  (Bool  "for" (Set (Var "M")) "," (Set (Var "b8")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Var "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b8"))  ($#r1_hidden :::"="::: )  (Set   ($#k40_descip_1 :::"DES-like-CoDec"::: )   "(" (Set (Var "M")) "," (Set (Var "F")) "," (Set (Var "IP")) "," (Set (Var "RK")) ")" )) "iff" (Bool  "ex" (Set (Var "L")) "," (Set (Var "R")) "being"   ($#m1_subset_1 :::"sequence":::) "of" (Set  "(" (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "st"  (Bool  "("  (Bool (Set (Set (Var "L"))  ($#k14_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k4_descip_1 :::"SP-Left"::: )  (Set  "(" (Set (Var "IP"))  ($#k3_funct_2 :::"."::: )  (Set (Var "M")) ")" ))) & (Bool (Set (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set  ($#k6_numbers :::"0"::: ) ))  ($#r1_hidden :::"="::: )  (Set   ($#k5_descip_1 :::"SP-Right"::: )  (Set  "(" (Set (Var "IP"))  ($#k3_funct_2 :::"."::: )  (Set (Var "M")) ")" ))) & (Bool  "("   "for" (Set (Var "i")) "being"    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  "st" (Bool (Bool (Set   ($#k6_numbers :::"0"::: )  )  ($#r1_xxreal_0 :::"<="::: )  (Set (Var "i"))) & (Bool (Set (Var "i"))  ($#r1_xxreal_0 :::"<="::: )  (Set (Set (Var "k"))  ($#k6_xcmplx_0 :::"-"::: )  (Num 1)))) "holds"  (Bool  "("  (Bool (Set (Set (Var "L"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")))) & (Bool (Set (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k12_descip_1 :::"Op-XOR"::: )   "(" (Set  "(" (Set (Var "L"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")) ")" ) "," (Set  "(" (Set (Var "F"))  ($#k2_binop_1 :::"."::: )   "(" (Set  "(" (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set (Var "i")) ")" ) "," (Set  "(" (Set (Var "RK"))  ($#k7_partfun1 :::"/."::: )  (Set  "(" (Set (Var "i"))  ($#k2_nat_1 :::"+"::: )  (Num 1) ")" ) ")" ) ")"  ")" ) ")" ))  ")" )  ")" ) & (Bool (Set (Var "b8"))  ($#r1_hidden :::"="::: )  (Set (Set  "(" (Set (Var "IP"))  ($#k2_funct_2 :::"""::: )  ")" )  ($#k1_funct_1 :::"."::: )  (Set  "(" (Set  "(" (Set (Var "R"))  ($#k14_descip_1 :::"."::: )  (Set (Var "k")) ")" )  ($#k9_binarith :::"^"::: )  (Set  "(" (Set (Var "L"))  ($#k14_descip_1 :::"."::: )  (Set (Var "k")) ")" ) ")" )))  ")" ))  ")" )))))); 
 
theorem :: DESCIP_1:38 (Bool  "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "k")) "being"  ($#~v1_xboole_0  "non"   ($#v1_xboole_0 :::"empty"::: )   )    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k5_numbers :::"NAT"::: )  )  (Bool  "for" (Set (Var "RK")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set (Var "k"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Set (Var "m"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ))  (Bool  "for" (Set (Var "F")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Set (Var "m"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set  "(" (Set (Var "n"))  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )  (Bool  "for" (Set (Var "IP")) "being"   ($#m1_subset_1 :::"Permutation":::) "of" (Set  "(" (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Var "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )  (Bool  "for" (Set (Var "M")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Set (Var "n")) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set   ($#k40_descip_1 :::"DES-like-CoDec"::: )   "(" (Set  "("  ($#k40_descip_1 :::"DES-like-CoDec"::: )   "(" (Set (Var "M")) "," (Set (Var "F")) "," (Set (Var "IP")) "," (Set (Var "RK")) ")"  ")" ) "," (Set (Var "F")) "," (Set (Var "IP")) "," (Set  "("  ($#k1_descip_1 :::"Rev"::: )  (Set (Var "RK")) ")" ) ")" )  ($#r1_hidden :::"="::: )  (Set (Var "M")))))))) ; 
 definitionlet "RK" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 16)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )); let "F" be   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ); let "IP" be   ($#m1_subset_1 :::"Permutation":::) "of" (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ); let "M" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-CoDec":::  "(" "M" "," "F" "," "IP" "," "RK" ")"  ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) means :: DESCIP_1:def 30 (Bool  "ex" (Set (Var "IPX")) "being"   ($#m1_subset_1 :::"Permutation":::) "of" (Set  "(" (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Num 32) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )(Bool  "ex" (Set (Var "MX")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Num 32) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "st"  (Bool  "("  (Bool (Set (Var "IPX"))  ($#r1_funct_2 :::"="::: )  "IP") & (Bool (Set (Var "MX"))  ($#r1_hidden :::"="::: )  "M") & (Bool it  ($#r1_hidden :::"="::: )  (Set   ($#k40_descip_1 :::"DES-like-CoDec"::: )   "(" (Set (Var "MX")) "," "F" "," (Set (Var "IPX")) "," "RK" ")" ))  ")" ))); end; 








:: deftheorem    defines :::"DES-CoDec"::: DESCIP_1:def 30 :  (Bool  "for" (Set (Var "RK")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 16)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ))  (Bool  "for" (Set (Var "F")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )  (Bool  "for" (Set (Var "IP")) "being"   ($#m1_subset_1 :::"Permutation":::) "of" (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )  (Bool  "for" (Set (Var "M")) "," (Set (Var "b5")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"   (Bool  "("  (Bool (Set (Var "b5"))  ($#r1_hidden :::"="::: )  (Set   ($#k41_descip_1 :::"DES-CoDec"::: )   "(" (Set (Var "M")) "," (Set (Var "F")) "," (Set (Var "IP")) "," (Set (Var "RK")) ")" )) "iff" (Bool  "ex" (Set (Var "IPX")) "being"   ($#m1_subset_1 :::"Permutation":::) "of" (Set  "(" (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Num 32) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )(Bool  "ex" (Set (Var "MX")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Set  "(" (Num 2)  ($#k4_nat_1 :::"*"::: )  (Num 32) ")" )  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "st"  (Bool  "("  (Bool (Set (Var "IPX"))  ($#r1_funct_2 :::"="::: )  (Set (Var "IP"))) & (Bool (Set (Var "MX"))  ($#r1_hidden :::"="::: )  (Set (Var "M"))) & (Bool (Set (Var "b5"))  ($#r1_hidden :::"="::: )  (Set   ($#k40_descip_1 :::"DES-like-CoDec"::: )   "(" (Set (Var "MX")) "," (Set (Var "F")) "," (Set (Var "IPX")) "," (Set (Var "RK")) ")" ))  ")" )))  ")" ))))); 
 
theorem :: DESCIP_1:39 (Bool  "for" (Set (Var "RK")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 16)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ))  (Bool  "for" (Set (Var "F")) "being"   ($#m1_subset_1 :::"Function":::) "of" (Set  ($#k2_zfmisc_1 :::"[:"::: ) (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set  "(" (Num 48)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set  "(" (Num 32)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )  (Bool  "for" (Set (Var "IP")) "being"   ($#m1_subset_1 :::"Permutation":::) "of" (Set  "(" (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )  (Bool  "for" (Set (Var "M")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set   ($#k41_descip_1 :::"DES-CoDec"::: )   "(" (Set  "("  ($#k41_descip_1 :::"DES-CoDec"::: )   "(" (Set (Var "M")) "," (Set (Var "F")) "," (Set (Var "IP")) "," (Set (Var "RK")) ")"  ")" ) "," (Set (Var "F")) "," (Set (Var "IP")) "," (Set  "("  ($#k1_descip_1 :::"Rev"::: )  (Set (Var "RK")) ")" ) ")" )  ($#r1_hidden :::"="::: )  (Set (Var "M"))))))) ; 
 definitionlet "plaintext", "secretkey" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-ENC":::  "(" "plaintext" "," "secretkey" ")"  ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) equals :: DESCIP_1:def 31 (Set   ($#k41_descip_1 :::"DES-CoDec"::: )   "(" "plaintext" "," (Set  ($#k35_descip_1 :::"DES-FFUNC"::: ) ) "," (Set  ($#k26_descip_1 :::"DES-PIP"::: ) ) "," (Set  "("  ($#k39_descip_1 :::"DES-KS"::: )  "secretkey" ")" ) ")" ); end; 






:: deftheorem    defines :::"DES-ENC"::: DESCIP_1:def 31 :  (Bool  "for" (Set (Var "plaintext")) "," (Set (Var "secretkey")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set   ($#k42_descip_1 :::"DES-ENC"::: )   "(" (Set (Var "plaintext")) "," (Set (Var "secretkey")) ")" )  ($#r1_hidden :::"="::: )  (Set   ($#k41_descip_1 :::"DES-CoDec"::: )   "(" (Set (Var "plaintext")) "," (Set  ($#k35_descip_1 :::"DES-FFUNC"::: ) ) "," (Set  ($#k26_descip_1 :::"DES-PIP"::: ) ) "," (Set  "("  ($#k39_descip_1 :::"DES-KS"::: )  (Set (Var "secretkey")) ")" ) ")" ))); 
 definitionlet "ciphertext", "secretkey" be    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )); func  :::"DES-DEC":::  "(" "ciphertext" "," "secretkey" ")"  ->    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) equals :: DESCIP_1:def 32 (Set   ($#k41_descip_1 :::"DES-CoDec"::: )   "(" "ciphertext" "," (Set  ($#k35_descip_1 :::"DES-FFUNC"::: ) ) "," (Set  ($#k26_descip_1 :::"DES-PIP"::: ) ) "," (Set  "("  ($#k1_descip_1 :::"Rev"::: )  (Set  "("  ($#k39_descip_1 :::"DES-KS"::: )  "secretkey" ")" ) ")" ) ")" ); end; 






:: deftheorem    defines :::"DES-DEC"::: DESCIP_1:def 32 :  (Bool  "for" (Set (Var "ciphertext")) "," (Set (Var "secretkey")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set   ($#k43_descip_1 :::"DES-DEC"::: )   "(" (Set (Var "ciphertext")) "," (Set (Var "secretkey")) ")" )  ($#r1_hidden :::"="::: )  (Set   ($#k41_descip_1 :::"DES-CoDec"::: )   "(" (Set (Var "ciphertext")) "," (Set  ($#k35_descip_1 :::"DES-FFUNC"::: ) ) "," (Set  ($#k26_descip_1 :::"DES-PIP"::: ) ) "," (Set  "("  ($#k1_descip_1 :::"Rev"::: )  (Set  "("  ($#k39_descip_1 :::"DES-KS"::: )  (Set (Var "secretkey")) ")" ) ")" ) ")" ))); 
 
theorem :: DESCIP_1:40 (Bool  "for" (Set (Var "message")) "," (Set (Var "secretkey")) "being"    ($#m2_finseq_2 :::"Element"::: )   "of" (Set (Num 64)  ($#k4_finseq_2 :::"-tuples_on"::: )  (Set  ($#k6_margrel1 :::"BOOLEAN"::: ) )) "holds"  (Bool (Set   ($#k43_descip_1 :::"DES-DEC"::: )   "(" (Set  "("  ($#k42_descip_1 :::"DES-ENC"::: )   "(" (Set (Var "message")) "," (Set (Var "secretkey")) ")"  ")" ) "," (Set (Var "secretkey")) ")" )  ($#r1_hidden :::"="::: )  (Set (Var "message")))) ;