:: RLAFFIN1 semantic presentation begin registrationlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "A" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "RLS")); cluster (Set ($#k3_convex1 :::"conv"::: ) "A") -> ($#v1_xboole_0 :::"empty"::: ) ; end; registrationlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "RLS")); cluster (Set ($#k3_convex1 :::"conv"::: ) "A") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: RLAFFIN1:1 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: RLAFFIN1:2 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "A")))))) ; theorem :: RLAFFIN1:3 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "B")))))) ; theorem :: RLAFFIN1:4 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "S"))))) "holds" (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" (Set (Var "S")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "A")) ")" ))))) ; theorem :: RLAFFIN1:5 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w")) ")" ) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_rusub_4 :::"+"::: ) (Set "(" (Set (Var "w")) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A")) ")" )))))) ; theorem :: RLAFFIN1:6 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))))) ; theorem :: RLAFFIN1:7 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "G")) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "g")) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A")) ")" )))))) ; theorem :: RLAFFIN1:8 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "v")) ($#k5_rusub_4 :::"+"::: ) (Set "(" ($#k1_struct_0 :::"{}"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "S")))))) ; theorem :: RLAFFIN1:9 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "B"))))))) ; theorem :: RLAFFIN1:10 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "AR")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set (Var "s")) ")" ) ($#k1_convex1 :::"*"::: ) (Set (Var "AR"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set "(" (Set (Var "s")) ($#k1_convex1 :::"*"::: ) (Set (Var "AR")) ")" )))))) ; theorem :: RLAFFIN1:11 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "AR")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Num 1) ($#k1_convex1 :::"*"::: ) (Set (Var "AR"))) ($#r1_hidden :::"="::: ) (Set (Var "AR"))))) ; theorem :: RLAFFIN1:12 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k1_convex1 :::"*"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: RLAFFIN1:13 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LS1")) "," (Set (Var "LS2")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set "(" (Set (Var "LS1")) ($#k7_rlvect_2 :::"+"::: ) (Set (Var "LS2")) ")" ) ($#k4_finseqop :::"*"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "LS1")) ($#k4_finseqop :::"*"::: ) (Set (Var "F")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set "(" (Set (Var "LS2")) ($#k4_finseqop :::"*"::: ) (Set (Var "F")) ")" )))))) ; theorem :: RLAFFIN1:14 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L")) ")" ) ($#k4_finseqop :::"*"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k10_rvsum_1 :::"*"::: ) (Set "(" (Set (Var "L")) ($#k4_finseqop :::"*"::: ) (Set (Var "F")) ")" ))))))) ; theorem :: RLAFFIN1:15 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "I")) "being" ($#v1_rlvect_3 :::"linearly-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set (Var "V"))) ")" )))) ; begin definitionlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "LG" be ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Const "G")); let "g" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "G")); func "g" :::"+"::: "LG" -> ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" "G" means :: RLAFFIN1:def 1 (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Element":::) "of" "G" "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set "LG" ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "h")) ($#k5_algstr_0 :::"-"::: ) "g" ")" )))); end; :: deftheorem defines :::"+"::: RLAFFIN1:def 1 : (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LG")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "b4")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "LG")))) "iff" (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set (Set (Var "b4")) ($#k1_seq_1 :::"."::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "LG")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "h")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "g")) ")" )))) ")" ))))); theorem :: RLAFFIN1:16 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LG")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set "(" (Set (Var "g")) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "LG")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k5_rusub_4 :::"+"::: ) (Set "(" ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "LG")) ")" )))))) ; theorem :: RLAFFIN1:17 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LG1")) "," (Set (Var "LG2")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set (Set (Var "g")) ($#k1_rlaffin1 :::"+"::: ) (Set "(" (Set (Var "LG1")) ($#k7_rlvect_2 :::"+"::: ) (Set (Var "LG2")) ")" )) ($#r1_rlvect_2 :::"="::: ) (Set (Set "(" (Set (Var "g")) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "LG1")) ")" ) ($#k7_rlvect_2 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "LG2")) ")" )))))) ; theorem :: RLAFFIN1:18 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k1_rlaffin1 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L")) ")" )) ($#r1_rlvect_2 :::"="::: ) (Set (Set (Var "r")) ($#k8_rlvect_2 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "L")) ")" ))))))) ; theorem :: RLAFFIN1:19 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LG")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set (Set (Var "g")) ($#k1_rlaffin1 :::"+"::: ) (Set "(" (Set (Var "h")) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "LG")) ")" )) ($#r1_rlvect_2 :::"="::: ) (Set (Set "(" (Set (Var "g")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "h")) ")" ) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "LG"))))))) ; theorem :: RLAFFIN1:20 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set (Set (Var "g")) ($#k1_rlaffin1 :::"+"::: ) (Set "(" ($#k4_rlvect_2 :::"ZeroLC"::: ) (Set (Var "G")) ")" )) ($#r1_rlvect_2 :::"="::: ) (Set ($#k4_rlvect_2 :::"ZeroLC"::: ) (Set (Var "G")))))) ; theorem :: RLAFFIN1:21 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LG")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "G")) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "G")) ")" ) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "LG"))) ($#r1_rlvect_2 :::"="::: ) (Set (Var "LG"))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "LR" be ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Const "R")); let "r" be ($#m1_subset_1 :::"Real":::); func "r" :::"(*)"::: "LR" -> ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" "R" means :: RLAFFIN1:def 2 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "R" "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set "LR" ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" "r" ($#k2_real_1 :::"""::: ) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" )))) if (Bool "r" ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) otherwise (Bool it ($#r1_rlvect_2 :::"="::: ) (Set ($#k4_rlvect_2 :::"ZeroLC"::: ) "R")); end; :: deftheorem defines :::"(*)"::: RLAFFIN1:def 2 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "LR")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "b4")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "R")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR")))) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "b4")) ($#k1_seq_1 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "LR")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "r")) ($#k2_real_1 :::"""::: ) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" )))) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )))) "implies" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR")))) "iff" (Bool (Set (Var "b4")) ($#r1_rlvect_2 :::"="::: ) (Set ($#k4_rlvect_2 :::"ZeroLC"::: ) (Set (Var "R")))) ")" ) ")" ")" ))))); theorem :: RLAFFIN1:22 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "LR")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "R")) "holds" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set "(" (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set "(" ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "LR")) ")" )))))) ; theorem :: RLAFFIN1:23 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "LR")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set "(" (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set "(" ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "LR")) ")" )))))) ; theorem :: RLAFFIN1:24 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "LR1")) "," (Set (Var "LR2")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set "(" (Set (Var "LR1")) ($#k7_rlvect_2 :::"+"::: ) (Set (Var "LR2")) ")" )) ($#r1_rlvect_2 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR1")) ")" ) ($#k7_rlvect_2 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR2")) ")" )))))) ; theorem :: RLAFFIN1:25 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "r")) ($#k8_rlvect_2 :::"*"::: ) (Set "(" (Set (Var "s")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "L")) ")" )) ($#r1_rlvect_2 :::"="::: ) (Set (Set (Var "s")) ($#k2_rlaffin1 :::"(*)"::: ) (Set "(" (Set (Var "r")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L")) ")" )))))) ; theorem :: RLAFFIN1:26 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) "holds" (Bool (Set (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set "(" ($#k4_rlvect_2 :::"ZeroLC"::: ) (Set (Var "R")) ")" )) ($#r1_rlvect_2 :::"="::: ) (Set ($#k4_rlvect_2 :::"ZeroLC"::: ) (Set (Var "R")))))) ; theorem :: RLAFFIN1:27 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "LR")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set "(" (Set (Var "s")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR")) ")" )) ($#r1_rlvect_2 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set (Var "s")) ")" ) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR"))))))) ; theorem :: RLAFFIN1:28 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "LR")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "R")) "holds" (Bool (Set (Num 1) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR"))) ($#r1_rlvect_2 :::"="::: ) (Set (Var "LR"))))) ; begin definitionlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "LS" be ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Const "S")); func :::"sum"::: "LS" -> ($#m1_subset_1 :::"Real":::) means :: RLAFFIN1:def 3 (Bool "ex" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" "S" "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) "LS")) & (Bool it ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" "LS" ($#k4_finseqop :::"*"::: ) (Set (Var "F")) ")" ))) ")" )); end; :: deftheorem defines :::"sum"::: RLAFFIN1:def 3 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LS")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "LS")))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "LS")))) & (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set (Var "LS")) ($#k4_finseqop :::"*"::: ) (Set (Var "F")) ")" ))) ")" )) ")" )))); theorem :: RLAFFIN1:29 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LS")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "LS"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set (Var "LS")) ($#k4_finseqop :::"*"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))))) ; theorem :: RLAFFIN1:30 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LS")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "LS"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "LS"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set (Var "LS")) ($#k4_finseqop :::"*"::: ) (Set (Var "F")) ")" )))))) ; theorem :: RLAFFIN1:31 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set "(" ($#k4_rlvect_2 :::"ZeroLC"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: RLAFFIN1:32 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LS")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "LS"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "LS"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "LS")) ($#k1_seq_1 :::"."::: ) (Set (Var "v"))))))) ; theorem :: RLAFFIN1:33 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LS")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "LS"))) ($#r1_tarski :::"c="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) )) & (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2")))) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "LS"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "LS")) ($#k1_seq_1 :::"."::: ) (Set (Var "v1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "LS")) ($#k1_seq_1 :::"."::: ) (Set (Var "v2")) ")" )))))) ; theorem :: RLAFFIN1:34 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LS1")) "," (Set (Var "LS2")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set "(" (Set (Var "LS1")) ($#k7_rlvect_2 :::"+"::: ) (Set (Var "LS2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "LS1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "LS2")) ")" ))))) ; theorem :: RLAFFIN1:35 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set "(" (Set (Var "r")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "L")) ")" )))))) ; theorem :: RLAFFIN1:36 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set "(" (Set (Var "L1")) ($#k10_rlvect_2 :::"-"::: ) (Set (Var "L2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "L1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "L2")) ")" ))))) ; theorem :: RLAFFIN1:37 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "LG")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "LG"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlaffin1 :::"sum"::: ) (Set "(" (Set (Var "g")) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "LG")) ")" )))))) ; theorem :: RLAFFIN1:38 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "LR")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "LR"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlaffin1 :::"sum"::: ) (Set "(" (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "LR")) ")" )))))) ; theorem :: RLAFFIN1:39 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set "(" (Set (Var "v")) ($#k1_rlaffin1 :::"+"::: ) (Set (Var "L")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "L")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L")) ")" )))))) ; theorem :: RLAFFIN1:40 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set "(" (Set (Var "r")) ($#k2_rlaffin1 :::"(*)"::: ) (Set (Var "L")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L")) ")" )))))) ; begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "A" is :::"affinely-independent"::: means :: RLAFFIN1:def 4 (Bool "(" (Bool "A" "is" ($#v1_xboole_0 :::"empty"::: ) ) "or" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" "st" (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "A") & (Bool (Set (Set "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_rusub_4 :::"+"::: ) "A" ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "V" ")" ) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) ")" )) ")" ); end; :: deftheorem defines :::"affinely-independent"::: RLAFFIN1:def 4 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) "iff" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_xboole_0 :::"empty"::: ) ) "or" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Set "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) ")" )) ")" ) ")" ))); registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v1_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"))); let "v" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "V")); cluster (Set ($#k1_tarski :::"{"::: ) "v" ($#k1_tarski :::"}"::: ) ) -> ($#v1_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "V"; let "w" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "V")); cluster (Set ($#k2_tarski :::"{"::: ) "v" "," "w" ($#k2_tarski :::"}"::: ) ) -> ($#v1_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "V"; end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster (Num 1) ($#v3_card_1 :::"-element"::: ) ($#v1_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"))); end; theorem :: RLAFFIN1:41 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) )) ")" ))) ; theorem :: RLAFFIN1:42 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) "iff" (Bool "for" (Set (Var "L")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" ))) ; theorem :: RLAFFIN1:43 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) & (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "B")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#v1_rlvect_3 :::"linearly-independent"::: ) -> ($#v1_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"))); end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "I" be ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); let "v" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "V")); cluster (Set "v" ($#k5_rusub_4 :::"+"::: ) "I") -> ($#v1_rlaffin1 :::"affinely-independent"::: ) ; end; theorem :: RLAFFIN1:44 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A"))) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "I" be ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); let "r" be ($#m1_subset_1 :::"Real":::); cluster (Set "r" ($#k1_convex1 :::"*"::: ) "I") -> ($#v1_rlaffin1 :::"affinely-independent"::: ) ; end; theorem :: RLAFFIN1:45 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "A"))) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )))) ; theorem :: RLAFFIN1:46 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) "iff" (Bool (Set (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) ")" ))) ; definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "F" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "V")); attr "F" is :::"affinely-independent"::: means :: RLAFFIN1:def 5 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "V" "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) "F")) "holds" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )); end; :: deftheorem defines :::"affinely-independent"::: RLAFFIN1:def 5 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_rlaffin1 :::"affinely-independent"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )) ")" ))); registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v2_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"))))); let "I" be ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); cluster (Set ($#k1_tarski :::"{"::: ) "I" ($#k1_tarski :::"}"::: ) ) -> ($#v2_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Subset-Family":::) "of" "V"; end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v2_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"))))); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"))))); end; theorem :: RLAFFIN1:47 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "F1")) "is" ($#v2_rlaffin1 :::"affinely-independent"::: ) ) & (Bool (Set (Var "F2")) "is" ($#v2_rlaffin1 :::"affinely-independent"::: ) )) "holds" (Bool (Set (Set (Var "F1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "F2"))) "is" ($#v2_rlaffin1 :::"affinely-independent"::: ) ))) ; theorem :: RLAFFIN1:48 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "F1")) ($#r1_tarski :::"c="::: ) (Set (Var "F2"))) & (Bool (Set (Var "F2")) "is" ($#v2_rlaffin1 :::"affinely-independent"::: ) )) "holds" (Bool (Set (Var "F1")) "is" ($#v2_rlaffin1 :::"affinely-independent"::: ) ))) ; begin definitionlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "RLS")); func :::"Affin"::: "A" -> ($#m1_subset_1 :::"Subset":::) "of" "RLS" equals :: RLAFFIN1:def 6 (Set ($#k1_setfam_1 :::"meet"::: ) "{" (Set (Var "B")) where B "is" ($#v2_rusub_4 :::"Affine"::: ) ($#m1_subset_1 :::"Subset":::) "of" "RLS" : (Bool "A" ($#r1_tarski :::"c="::: ) (Set (Var "B"))) "}" ); end; :: deftheorem defines :::"Affin"::: RLAFFIN1:def 6 : (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) "{" (Set (Var "B")) where B "is" ($#v2_rusub_4 :::"Affine"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) : (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) "}" )))); registrationlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "RLS")); cluster (Set ($#k4_rlaffin1 :::"Affin"::: ) "A") -> ($#v2_rusub_4 :::"Affine"::: ) ; end; registrationlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "A" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "RLS")); cluster (Set ($#k4_rlaffin1 :::"Affin"::: ) "A") -> ($#v1_xboole_0 :::"empty"::: ) ; end; registrationlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "RLS")); cluster (Set ($#k4_rlaffin1 :::"Affin"::: ) "A") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: RLAFFIN1:49 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))))) ; theorem :: RLAFFIN1:50 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#v2_rusub_4 :::"Affine"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))))) ; theorem :: RLAFFIN1:51 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Var "B")) "is" ($#v2_rusub_4 :::"Affine"::: ) )) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Var "B"))))) ; theorem :: RLAFFIN1:52 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B")))))) ; theorem :: RLAFFIN1:53 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set "(" (Set (Var "v")) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_rusub_4 :::"+"::: ) (Set "(" ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")) ")" )))))) ; theorem :: RLAFFIN1:54 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "AR")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "AR")) "is" ($#v2_rusub_4 :::"Affine"::: ) )) "holds" (Bool (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "AR"))) "is" ($#v2_rusub_4 :::"Affine"::: ) )))) ; theorem :: RLAFFIN1:55 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "AR")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set "(" (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "AR")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set "(" ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "AR")) ")" )))))) ; theorem :: RLAFFIN1:56 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set "(" (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set "(" ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")) ")" )))))) ; theorem :: RLAFFIN1:57 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_rusub_4 :::"+"::: ) (Set "(" ($#k3_rusub_4 :::"Up"::: ) (Set "(" ($#k1_rlvect_3 :::"Lin"::: ) (Set "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_rusub_4 :::"+"::: ) (Set (Var "A")) ")" ) ")" ) ")" )))))) ; theorem :: RLAFFIN1:58 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) "iff" (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) ")" ))) ; theorem :: RLAFFIN1:59 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L")) ")" ) where L "is" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "A")) : (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Num 1)) "}" ))) ; theorem :: RLAFFIN1:60 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "Ia")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Var "Ia"))) & (Bool (Set (Var "Ia")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "Ia"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))) ")" ))))) ; theorem :: RLAFFIN1:61 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) & (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "B"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Var "B")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ))) ; theorem :: RLAFFIN1:62 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_convex1 :::"convex"::: ) ) "iff" (Bool "(" (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set (Var "v")))) ")" ) ")" ) ")" ))) ; theorem :: RLAFFIN1:63 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v2_convex1 :::"convex"::: ) )) "holds" (Bool (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1))))) ; theorem :: RLAFFIN1:64 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v2_convex1 :::"convex"::: ) ) & (Bool (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: RLAFFIN1:65 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))))) ; theorem :: RLAFFIN1:66 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "A")))) & (Bool (Set (Set "(" ($#k3_convex1 :::"conv"::: ) (Set (Var "A")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) "is" ($#v1_convex1 :::"convex"::: ) )) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))))) ; theorem :: RLAFFIN1:67 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set "(" ($#k3_convex1 :::"conv"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))))) ; theorem :: RLAFFIN1:68 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "B"))))) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B")))))) ; theorem :: RLAFFIN1:69 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B"))))) "holds" (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B")))))) ; begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); assume (Bool (Set (Const "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) ; let "x" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Const "A")))) ; func "x" :::"|--"::: "A" -> ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" "A" means :: RLAFFIN1:def 7 (Bool "(" (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) it) ($#r1_hidden :::"="::: ) "x") & (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) it) ($#r1_hidden :::"="::: ) (Num 1)) ")" ); end; :: deftheorem defines :::"|--"::: RLAFFIN1:def 7 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))))) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "A")))) "iff" (Bool "(" (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k3_rlaffin1 :::"sum"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ))))); theorem :: RLAFFIN1:70 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "I")))) & (Bool (Set (Var "v2")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "I"))))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" ) ")" ) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I"))) ($#r1_rlvect_2 :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k8_rlvect_2 :::"*"::: ) (Set "(" (Set (Var "v1")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ")" ) ($#k7_rlvect_2 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k8_rlvect_2 :::"*"::: ) (Set "(" (Set (Var "v2")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ")" ))))))) ; theorem :: RLAFFIN1:71 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "I"))))) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I"))) "is" ($#v2_convex1 :::"convex"::: ) ) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "v")))) & (Bool (Set (Set "(" (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "v"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: RLAFFIN1:72 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "I"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) ")" ) ")" )))) ; theorem :: RLAFFIN1:73 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "I")))) & (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "v")))) ")" )) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "I"))))))) ; theorem :: RLAFFIN1:74 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set (Set "(" ($#k3_convex1 :::"conv"::: ) (Set (Var "I")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) "is" ($#v1_convex1 :::"convex"::: ) )))) ; theorem :: RLAFFIN1:75 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "I")))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set "(" (Set (Var "I")) ($#k7_subset_1 :::"\"::: ) (Set (Var "B")) ")" ))) & (Bool (Set (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I"))) ($#r1_rlvect_2 :::"="::: ) (Set (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set "(" (Set (Var "I")) ($#k7_subset_1 :::"\"::: ) (Set (Var "B")) ")" ))) ")" ))))) ; theorem :: RLAFFIN1:76 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "I")))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" (Set (Var "I")) ($#k7_subset_1 :::"\"::: ) (Set (Var "B")) ")" ))))))) ; theorem :: RLAFFIN1:77 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "I"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "B"))) ($#r1_rlvect_2 :::"="::: ) (Set (Set (Var "x")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")))))))) ; theorem :: RLAFFIN1:78 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))) & (Bool (Set (Var "v2")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))) & (Bool (Set (Set (Var "r")) ($#k7_real_1 :::"+"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "s")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))))))) ; theorem :: RLAFFIN1:79 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) & (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B"))))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "B")))))) ; theorem :: RLAFFIN1:80 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) & (Bool (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "B")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Var "B")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ))) ; theorem :: RLAFFIN1:81 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "L1")) ($#k1_seq_1 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "L2")) ($#k1_seq_1 :::"."::: ) (Set (Var "v"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Set (Var "r")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L1")) ")" ) ($#k7_rlvect_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L2")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Var "s"))) "iff" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "L2")) ($#k1_seq_1 :::"."::: ) (Set (Var "v")) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "L2")) ($#k1_seq_1 :::"."::: ) (Set (Var "v")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "L1")) ($#k1_seq_1 :::"."::: ) (Set (Var "v")) ")" ) ")" ))) ")" ))))) ; theorem :: RLAFFIN1:82 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) "iff" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) & (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "or" "not" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A")))) ")" ) ")" ) ")" )))) ; theorem :: RLAFFIN1:83 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "w")) "," (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Bool "not" (Set (Var "w")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "A"))))) & (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "v2")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" )))) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool (Set (Var "v1")) ($#r1_hidden :::"="::: ) (Set (Var "v2"))) ")" ))))) ; theorem :: RLAFFIN1:84 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "I")) "being" ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "w")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set (Var "I")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rlaffin1 :::"Affin"::: ) (Set "(" (Set (Var "I")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) ")" ))) & (Bool (Set (Var "w")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "w")) ($#k5_rlaffin1 :::"|--"::: ) (Set (Var "I")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "v")))))))) ;