:: VECTSP_1 semantic presentation begin definitionfunc :::"G_Real"::: -> ($#v8_algstr_0 :::"strict"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) equals :: VECTSP_1:def 1 (Set ($#g2_algstr_0 :::"addLoopStr"::: ) "(#" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k33_binop_2 :::"addreal"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "#)" ); end; :: deftheorem defines :::"G_Real"::: VECTSP_1:def 1 : (Bool (Set ($#k1_vectsp_1 :::"G_Real"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#g2_algstr_0 :::"addLoopStr"::: ) "(#" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k33_binop_2 :::"addreal"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "#)" )); registration cluster (Set ($#k1_vectsp_1 :::"G_Real"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_algstr_0 :::"strict"::: ) ; end; registration cluster (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k1_vectsp_1 :::"G_Real"::: ) )) -> ($#v3_membered :::"real-membered"::: ) ; end; registrationlet "a", "b" be ($#m1_subset_1 :::"Element":::) "of" (Set ($#k1_vectsp_1 :::"G_Real"::: ) ); let "x", "y" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; identify ; end; registration cluster (Set ($#k1_vectsp_1 :::"G_Real"::: ) ) -> ($#v8_algstr_0 :::"strict"::: ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ; end; registrationlet "a" be ($#m1_subset_1 :::"Element":::) "of" (Set ($#k1_vectsp_1 :::"G_Real"::: ) ); let "x" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; identify ; end; theorem :: VECTSP_1:1 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k1_vectsp_1 :::"G_Real"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set "(" (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "y")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "y")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "z")) ")" ))) & (Bool (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set ($#k1_vectsp_1 :::"G_Real"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k1_vectsp_1 :::"G_Real"::: ) ))) ")" )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_algstr_0 :::"strict"::: ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) for ($#l2_algstr_0 :::"addLoopStr"::: ) ; end; definitionmode AddGroup is ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; end; definitionmode AbGroup is ($#v2_rlvect_1 :::"Abelian"::: ) ($#l2_algstr_0 :::"AddGroup":::); end; definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; attr "IT" is :::"right-distributive"::: means :: VECTSP_1:def 2 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "b")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "c")) ")" )))); attr "IT" is :::"left-distributive"::: means :: VECTSP_1:def 3 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set "(" (Set (Var "b")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "b")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "c")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a")) ")" )))); end; :: deftheorem defines :::"right-distributive"::: VECTSP_1:def 2 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_vectsp_1 :::"right-distributive"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "b")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "c")) ")" )))) ")" )); :: deftheorem defines :::"left-distributive"::: VECTSP_1:def 3 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_vectsp_1 :::"left-distributive"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" (Set (Var "b")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "b")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "c")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a")) ")" )))) ")" )); definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; attr "IT" is :::"right_unital"::: means :: VECTSP_1:def 4 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) "IT" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x")))); end; :: deftheorem defines :::"right_unital"::: VECTSP_1:def 4 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v3_vectsp_1 :::"right_unital"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "IT")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x")))) ")" )); definitionfunc :::"F_Real"::: -> ($#v36_algstr_0 :::"strict"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) equals :: VECTSP_1:def 5 (Set ($#g6_algstr_0 :::"doubleLoopStr"::: ) "(#" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k33_binop_2 :::"addreal"::: ) ) "," (Set ($#k35_binop_2 :::"multreal"::: ) ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) "#)" ); end; :: deftheorem defines :::"F_Real"::: VECTSP_1:def 5 : (Bool (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#g6_algstr_0 :::"doubleLoopStr"::: ) "(#" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k33_binop_2 :::"addreal"::: ) ) "," (Set ($#k35_binop_2 :::"multreal"::: ) ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) "#)" )); registration cluster (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ; end; registration cluster (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) )) -> ($#v3_membered :::"real-membered"::: ) ; end; registrationlet "a", "b" be ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ); let "x", "y" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; identify ; end; registrationlet "a", "b" be ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ); let "x", "y" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; identify ; end; definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; attr "IT" is :::"well-unital"::: means :: VECTSP_1:def 6 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) "IT" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) "IT" ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" )); end; :: deftheorem defines :::"well-unital"::: VECTSP_1:def 6 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v4_vectsp_1 :::"well-unital"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "IT")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "IT")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" )) ")" )); registration cluster (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) -> ($#v36_algstr_0 :::"strict"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) for ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; end; definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; attr "IT" is :::"distributive"::: means :: VECTSP_1:def 7 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "z")) ")" ))) & (Bool (Set (Set "(" (Set (Var "y")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "z")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x")) ")" ))) ")" )); end; :: deftheorem defines :::"distributive"::: VECTSP_1:def 7 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v5_vectsp_1 :::"distributive"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "z")) ")" ))) & (Bool (Set (Set "(" (Set (Var "y")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "z")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x")) ")" ))) ")" )) ")" )); definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; attr "IT" is :::"left_unital"::: means :: VECTSP_1:def 8 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) "IT" ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))); end; :: deftheorem defines :::"left_unital"::: VECTSP_1:def 8 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v6_vectsp_1 :::"left_unital"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "IT")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))) ")" )); definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; redefine attr "IT" is :::"almost_left_invertible"::: means :: VECTSP_1:def 9 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "IT"))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "st" (Bool (Set (Set (Var "y")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) "IT")))); end; :: deftheorem defines :::"almost_left_invertible"::: VECTSP_1:def 9 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v33_algstr_0 :::"almost_left_invertible"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "IT"))))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "st" (Bool (Set (Set (Var "y")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "IT")))))) ")" )); registration cluster (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) -> ($#v36_algstr_0 :::"strict"::: ) ($#v1_group_1 :::"unital"::: ) ; end; registration cluster (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) -> ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v36_algstr_0 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) ; end; registrationlet "a" be ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ); let "x" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; identify ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_vectsp_1 :::"distributive"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_vectsp_1 :::"distributive"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) for ($#l4_algstr_0 :::"multLoopStr"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_group_1 :::"unital"::: ) for ($#l4_algstr_0 :::"multLoopStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) for ($#l3_algstr_0 :::"multMagma"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_group_1 :::"unital"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) for ($#l4_algstr_0 :::"multLoopStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v36_algstr_0 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; definitionmode Field is ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; theorem :: VECTSP_1:2 (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) ; theorem :: VECTSP_1:3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set "(" (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "y")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "y")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "z")) ")" ))) & (Bool (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) ))) & (Bool (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k8_group_1 :::"*"::: ) (Set (Var "x")))) & (Bool (Set (Set "(" (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set (Var "y")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k8_group_1 :::"*"::: ) (Set (Var "z")) ")" ))) & (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & "(" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) )))) "implies" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Set (Set (Var "y")) ($#k8_group_1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) )))) ")" & (Bool (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set (Var "y")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set (Var "z")) ")" ))) & (Bool (Set (Set "(" (Set (Var "y")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "z")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k8_group_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k8_group_1 :::"*"::: ) (Set (Var "x")) ")" ))) ")" )) ; theorem :: VECTSP_1:4 (Bool "for" (Set (Var "FS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "FS")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "FS"))))) "implies" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "FS")) "st" (Bool (Set (Set (Var "y")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "FS"))))) ")" & (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "z")) ")" ))) & (Bool (Set (Set "(" (Set (Var "y")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "z")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x")) ")" ))) ")" ) ")" ) "iff" (Bool (Set (Var "FS")) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ) ")" )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_group_1 :::"unital"::: ) for ($#l4_algstr_0 :::"multLoopStr"::: ) ; end; theorem :: VECTSP_1:5 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) & (Bool (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set (Var "z"))))) "holds" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "z"))))) ; definitionlet "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "F")); assume (Bool (Set (Const "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Const "F")))) ; redefine func "x" :::"""::: means :: VECTSP_1:def 10 (Bool (Set it ($#k8_group_1 :::"*"::: ) "x") ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) "F")); end; :: deftheorem defines :::"""::: VECTSP_1:def 10 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "F"))) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k11_algstr_0 :::"""::: ) )) "iff" (Bool (Set (Set (Var "b3")) ($#k8_group_1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "F")))) ")" )))); definitionlet "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "x", "y" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "F")); func "x" :::"/"::: "y" -> ($#m1_subset_1 :::"Element":::) "of" "F" equals :: VECTSP_1:def 11 (Set "x" ($#k8_group_1 :::"*"::: ) (Set "(" "y" ($#k11_algstr_0 :::"""::: ) ")" )); end; :: deftheorem defines :::"/"::: VECTSP_1:def 11 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set (Var "x")) ($#k3_vectsp_1 :::"/"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k11_algstr_0 :::"""::: ) ")" ))))); theorem :: VECTSP_1:6 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))))) ; theorem :: VECTSP_1:7 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "F")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))))) ; theorem :: VECTSP_1:8 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")) ")" ))))) ; theorem :: VECTSP_1:9 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "x")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")) ")" ))))) ; theorem :: VECTSP_1:10 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "x")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")))))) ; theorem :: VECTSP_1:11 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "z")) ")" ))))) ; theorem :: VECTSP_1:12 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) "or" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) ")" ) ")" ))) ; theorem :: VECTSP_1:13 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "b")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "c")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "b")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "c")) ")" ))))) ; definitionlet "F" be ($#l1_struct_0 :::"1-sorted"::: ) ; attr "c2" is :::"strict"::: ; struct :::"VectSpStr"::: "over" "F" -> ($#l2_algstr_0 :::"addLoopStr"::: ) ; aggr :::"VectSpStr":::(# :::"carrier":::, :::"addF":::, :::"ZeroF":::, :::"lmult"::: #) -> ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" "F"; sel :::"lmult"::: "c2" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "F") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c2") ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c2"); end; registrationlet "F" be ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_vectsp_1 :::"strict"::: ) for ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" "F"; end; registrationlet "F" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "a" be ($#m1_subset_1 :::"BinOp":::) "of" (Set (Const "A")); let "Z" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "A")); let "l" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "F"))) "," (Set (Const "A")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Const "A")); cluster (Set ($#g1_vectsp_1 :::"VectSpStr"::: ) "(#" "A" "," "a" "," "Z" "," "l" "#)" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; definitionlet "F" be ($#l1_struct_0 :::"1-sorted"::: ) ; mode Scalar of "F" is ($#m1_subset_1 :::"Element":::) "of" "F"; end; definitionlet "F" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "VS" be ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Const "F")); mode Scalar of "VS" is ($#m1_subset_1 :::"Scalar":::) "of" "F"; mode Vector of "VS" is ($#m1_subset_1 :::"Element":::) "of" "VS"; end; definitionlet "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Const "F")); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "F")); let "v" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "V")); func "x" :::"*"::: "v" -> ($#m1_subset_1 :::"Element":::) "of" "V" equals :: VECTSP_1:def 12 (Set (Set "the" ($#u1_vectsp_1 :::"lmult"::: ) "of" "V") ($#k2_binop_1 :::"."::: ) "(" "x" "," "v" ")" ); end; :: deftheorem defines :::"*"::: VECTSP_1:def 12 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_vectsp_1 :::"lmult"::: ) "of" (Set (Var "V"))) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "v")) ")" )))))); definitionlet "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; func :::"comp"::: "F" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "F") means :: VECTSP_1:def 13 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "F" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "x"))))); end; :: deftheorem defines :::"comp"::: VECTSP_1:def 13 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "F"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_vectsp_1 :::"comp"::: ) (Set (Var "F")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "x"))))) ")" ))); definitionlet "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Const "F")); attr "IT" is :::"vector-distributive"::: means :: VECTSP_1:def 14 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "F" (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "w")) ")" ))))); attr "IT" is :::"scalar-distributive"::: means :: VECTSP_1:def 15 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "F" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "y")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ))))); attr "IT" is :::"scalar-associative"::: means :: VECTSP_1:def 16 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "F" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ))))); attr "IT" is :::"scalar-unital"::: means :: VECTSP_1:def 17 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "holds" (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) "F" ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Var "v")))); end; :: deftheorem defines :::"vector-distributive"::: VECTSP_1:def 14 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v8_vectsp_1 :::"vector-distributive"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "w")) ")" ))))) ")" ))); :: deftheorem defines :::"scalar-distributive"::: VECTSP_1:def 15 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v9_vectsp_1 :::"scalar-distributive"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "y")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ))))) ")" ))); :: deftheorem defines :::"scalar-associative"::: VECTSP_1:def 16 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v10_vectsp_1 :::"scalar-associative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ))))) ")" ))); :: deftheorem defines :::"scalar-unital"::: VECTSP_1:def 17 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v11_vectsp_1 :::"scalar-unital"::: ) ) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "F")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Var "v")))) ")" ))); registrationlet "F" 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"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v7_vectsp_1 :::"strict"::: ) ($#v8_vectsp_1 :::"vector-distributive"::: ) ($#v9_vectsp_1 :::"scalar-distributive"::: ) ($#v10_vectsp_1 :::"scalar-associative"::: ) ($#v11_vectsp_1 :::"scalar-unital"::: ) for ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" "F"; end; definitionlet "F" 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"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; mode VectSp of "F" is ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v8_vectsp_1 :::"vector-distributive"::: ) ($#v9_vectsp_1 :::"scalar-distributive"::: ) ($#v10_vectsp_1 :::"scalar-associative"::: ) ($#v11_vectsp_1 :::"scalar-unital"::: ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" "F"; end; theorem :: VECTSP_1:14 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v8_vectsp_1 :::"vector-distributive"::: ) ($#v9_vectsp_1 :::"scalar-distributive"::: ) ($#v10_vectsp_1 :::"scalar-associative"::: ) ($#v11_vectsp_1 :::"scalar-unital"::: ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "F")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "F")) ")" ) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")))) & (Bool (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) ")" ))))) ; theorem :: VECTSP_1:15 (Bool "for" (Set (Var "F")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) "or" (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) ")" ) ")" ))))) ; theorem :: VECTSP_1:16 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) "iff" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Var "w"))) ")" ))) ; theorem :: VECTSP_1:17 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "w")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v")))) & (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "w")) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")))) & (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "w")) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ))) & (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "w")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v")))) & (Bool (Set (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "w")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")))) ")" ))) ; theorem :: VECTSP_1:18 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")))) & (Bool (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "v"))) ")" ))) ; theorem :: VECTSP_1:19 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "implies" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" & "(" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "implies" (Bool (Set (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) ")" & "(" (Bool (Bool (Set (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "implies" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" & "(" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "implies" (Bool (Set (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) ")" ")" ))) ; theorem :: VECTSP_1:20 (Bool "for" (Set (Var "F")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "v"))))))) ; theorem :: VECTSP_1:21 (Bool "for" (Set (Var "F")) "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"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v8_vectsp_1 :::"vector-distributive"::: ) ($#v9_vectsp_1 :::"scalar-distributive"::: ) ($#v10_vectsp_1 :::"scalar-associative"::: ) ($#v11_vectsp_1 :::"scalar-unital"::: ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "x")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")))) & (Bool (Set (Set (Var "w")) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "w")) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "x")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ))) ")" ))))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_vectsp_1 :::"right_unital"::: ) for ($#l4_algstr_0 :::"multLoopStr"::: ) ; end; theorem :: VECTSP_1:22 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v8_vectsp_1 :::"vector-distributive"::: ) ($#v9_vectsp_1 :::"scalar-distributive"::: ) ($#v10_vectsp_1 :::"scalar-associative"::: ) ($#v11_vectsp_1 :::"scalar-unital"::: ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ))))))) ; theorem :: VECTSP_1:23 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v8_vectsp_1 :::"vector-distributive"::: ) ($#v9_vectsp_1 :::"scalar-distributive"::: ) ($#v10_vectsp_1 :::"scalar-associative"::: ) ($#v11_vectsp_1 :::"scalar-unital"::: ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set (Var "F")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "w")) ")" ))))))) ; theorem :: VECTSP_1:24 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k11_algstr_0 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: VECTSP_1:25 (Bool "for" (Set (Var "F")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool "(" (Bool (Set (Set (Var "x")) ($#k11_algstr_0 :::"""::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) & (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k11_algstr_0 :::"""::: ) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) ")" ))) ; theorem :: VECTSP_1:26 (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_vectsp_1 :::"F_Real"::: ) ))) ; definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; attr "IT" is :::"Fanoian"::: means :: VECTSP_1:def 18 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "IT" "st" (Bool (Bool (Set (Set (Var "a")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "IT"))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "IT"))); end; :: deftheorem defines :::"Fanoian"::: VECTSP_1:def 18 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v12_vectsp_1 :::"Fanoian"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "IT")) "st" (Bool (Bool (Set (Set (Var "a")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "IT"))))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "IT"))))) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_vectsp_1 :::"Fanoian"::: ) for ($#l2_algstr_0 :::"addLoopStr"::: ) ; end; definitionlet "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; redefine attr "F" is :::"Fanoian"::: means :: VECTSP_1:def 19 (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) "F" ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) "F" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "F")); end; :: deftheorem defines :::"Fanoian"::: VECTSP_1:def 19 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v12_vectsp_1 :::"Fanoian"::: ) ) "iff" (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "F")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v36_algstr_0 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_group_1 :::"unital"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#v12_vectsp_1 :::"Fanoian"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; theorem :: VECTSP_1:27 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set (Set (Var "a")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b"))))) ; theorem :: VECTSP_1:28 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))))) ; theorem :: VECTSP_1:29 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set (Set (Var "a")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "b")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))))) ; theorem :: VECTSP_1:30 (Bool "for" (Set (Var "F")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) & (Bool (Set (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "c")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "implies" (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k11_algstr_0 :::"""::: ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) & (Bool (Set (Set (Var "b")) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "c")) ($#k8_group_1 :::"*"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "implies" (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k11_algstr_0 :::"""::: ) ")" ))) ")" ")" ))) ; theorem :: VECTSP_1:31 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set (Var "a")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "b")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ))))) ; theorem :: VECTSP_1:32 (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set "(" (Set (Var "b")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "c")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "c")))))) ; theorem :: VECTSP_1:33 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "w")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v")))))) ; theorem :: VECTSP_1:34 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set (Set "(" (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v")))))) ; theorem :: VECTSP_1:35 (Bool "for" (Set (Var "B")) "being" ($#l2_algstr_0 :::"AbGroup":::) "holds" (Bool (Set ($#g3_algstr_0 :::"multMagma"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "B"))) "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "B"))) "#)" ) "is" ($#v5_group_1 :::"commutative"::: ) ($#l3_algstr_0 :::"Group":::))) ; begin theorem :: VECTSP_1:36 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_group_1 :::"unital"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "L")) ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "L")))))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) for ($#l4_algstr_0 :::"multLoopStr"::: ) ; end; registrationlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; identify ; end; theorem :: VECTSP_1:37 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) "st" (Bool (Bool (Set (Var "L")) "is" ($#v4_vectsp_1 :::"well-unital"::: ) )) "holds" (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k1_group_1 :::"1_"::: ) (Set (Var "L"))))) ; definitionlet "G", "H" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_algstr_0 :::"addMagma"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "G")) "," (Set (Const "H")); attr "f" is :::"additive"::: means :: VECTSP_1:def 20 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "G" "holds" (Bool (Set "f" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" )))); end; :: deftheorem defines :::"additive"::: VECTSP_1:def 20 : (Bool "for" (Set (Var "G")) "," (Set (Var "H")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_algstr_0 :::"addMagma"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "G")) "," (Set (Var "H")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v13_vectsp_1 :::"additive"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" )))) ")" )));