:: RLVECT_5 semantic presentation begin theorem :: RLVECT_5:1 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "KL1")) "," (Set (Var "KL2")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "KL1"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "KL2"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "KL1"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "KL2"))))) "holds" (Bool (Set (Var "KL1")) ($#r1_rlvect_2 :::"="::: ) (Set (Var "KL2")))))) ; theorem :: RLVECT_5:2 (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" ($#v1_rlvect_3 :::"linearly-independent"::: ) )) "holds" (Bool "ex" (Set (Var "I")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "I")))))) ; theorem :: RLVECT_5:3 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L")))) "iff" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" )))) ; theorem :: RLVECT_5:4 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set (Var "F")) ")" ) "st" (Bool (Bool (Set (Var "G")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k1_partfun1 :::"*"::: ) (Set (Var "P"))))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "L")) ($#k5_rlvect_2 :::"(#)"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "L")) ($#k5_rlvect_2 :::"(#)"::: ) (Set (Var "G")) ")" ))))))) ; theorem :: RLVECT_5:5 (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" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "L")) ($#k5_rlvect_2 :::"(#)"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))))) ; theorem :: RLVECT_5:6 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "L")) ($#k5_rlvect_2 :::"(#)"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))))))) ; theorem :: RLVECT_5:7 (Bool "for" (Set (Var "V")) "being" ($#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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "ex" (Set (Var "K")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L")) ")" ))) & (Bool (Set (Set (Var "L")) ($#k5_rlvect_2 :::"(#)"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "K")) ($#k5_rlvect_2 :::"(#)"::: ) (Set (Var "F")))) ")" ))))) ; theorem :: RLVECT_5:8 (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 "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "A")) ")" )))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "st" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "L")) ($#k5_rlvect_2 :::"(#)"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "K"))))))))) ; theorem :: RLVECT_5:9 (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 "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "A")) ")" )))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "st" (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "K")))))))) ; theorem :: RLVECT_5:10 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))) "holds" (Bool "for" (Set (Var "K")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "K")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k2_partfun1 :::"|"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))))) "holds" (Bool "(" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "K")))) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "K")))) ")" ))))) ; theorem :: RLVECT_5:11 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "K")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "W")) (Bool "ex" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L")))) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L")))) ")" ))))) ; theorem :: RLVECT_5:12 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "W")) "st" (Bool "(" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L")))) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L")))) ")" ))))) ; theorem :: RLVECT_5:13 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "I")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "I"))))))) ; theorem :: RLVECT_5:14 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")))))) ; theorem :: RLVECT_5:15 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))) "holds" (Bool (Set (Var "A")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "W")))))) ; theorem :: RLVECT_5:16 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "W")) (Bool "ex" (Set (Var "B")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))))))) ; theorem :: RLVECT_5:17 (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" ($#v1_rlvect_3 :::"linearly-independent"::: ) )) "holds" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )))) "holds" (Bool "not" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "B"))))))))) ; theorem :: RLVECT_5:18 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "I")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "I")))) "holds" (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Var "B")) "is" ($#v1_rlvect_3 :::"linearly-dependent"::: ) ))))) ; theorem :: RLVECT_5:19 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))) "holds" (Bool (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "A"))) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "W")))))) ; theorem :: RLVECT_5:20 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "B")))))))) ; begin theorem :: RLVECT_5:21 (Bool "for" (Set (Var "V")) "being" ($#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")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ))) & (Bool (Bool "not" (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "B")))))) "holds" (Bool "ex" (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "w")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "w")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "w")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) ")" ))) ")" ))))) ; theorem :: RLVECT_5:22 (Bool "for" (Set (Var "V")) "being" ($#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 ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_algstr_0 :::"U5"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "A")))) & (Bool (Set (Var "B")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) )) "holds" (Bool "(" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "B"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "A")))) & (Bool "ex" (Set (Var "C")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "A")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "B")) ")" ))) & (Bool (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_algstr_0 :::"U5"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set "(" (Set (Var "B")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "C")) ")" ))) ")" )) ")" ))) ; begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); attr "V" is :::"finite-dimensional"::: means :: RLVECT_5:def 1 (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" "V" "st" (Bool (Set (Var "A")) "is" ($#m1_rlvect_3 :::"Basis"::: ) "of" "V")); end; :: deftheorem defines :::"finite-dimensional"::: RLVECT_5:def 1 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool "(" (Bool (Set (Var "V")) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ) "iff" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Set (Var "A")) "is" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "V")))) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) bbbadV13_ALGSTR_0() ($#v1_rlvect_1 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#v1_rlvect_5 :::"finite-dimensional"::: ) for ($#l1_rlvect_1 :::"RLSStruct"::: ) ; end; theorem :: RLVECT_5:23 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) )) "holds" (Bool "for" (Set (Var "I")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Var "I")) "is" ($#v1_finset_1 :::"finite"::: ) ))) ; theorem :: RLVECT_5:24 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) )) "holds" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"finite"::: ) ))) ; theorem :: RLVECT_5:25 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) )) "holds" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "B")))))) ; theorem :: RLVECT_5:26 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V"))) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) )) ; theorem :: RLVECT_5:27 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V")) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) )) "holds" (Bool (Set (Var "W")) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) bbbadV13_ALGSTR_0() ($#v1_rlvect_1 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#v1_rlvect_5 :::"finite-dimensional"::: ) for ($#m1_rlsub_1 :::"Subspace"::: ) "of" "V"; end; registrationlet "V" be ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster -> ($#v1_rlvect_5 :::"finite-dimensional"::: ) for ($#m1_rlsub_1 :::"Subspace"::: ) "of" "V"; end; registrationlet "V" be ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) bbbadV13_ALGSTR_0() ($#v1_rlvect_1 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#v1_rlvect_5 :::"finite-dimensional"::: ) for ($#m1_rlsub_1 :::"Subspace"::: ) "of" "V"; end; begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); assume (Bool (Set (Const "V")) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ) ; func :::"dim"::: "V" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: RLVECT_5:def 2 (Bool "for" (Set (Var "I")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" "V" "holds" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "I"))))); end; :: deftheorem defines :::"dim"::: RLVECT_5:def 2 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#v1_rlvect_5 :::"finite-dimensional"::: ) )) "holds" (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V")))) "iff" (Bool "for" (Set (Var "I")) "being" ($#m1_rlvect_3 :::"Basis"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "I"))))) ")" ))); theorem :: RLVECT_5:28 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V")))))) ; theorem :: RLVECT_5:29 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) )) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_5 :::"dim"::: ) (Set "(" ($#k1_rlvect_3 :::"Lin"::: ) (Set (Var "A")) ")" ))))) ; theorem :: RLVECT_5:30 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_5 :::"dim"::: ) (Set "(" ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V")) ")" )))) ; theorem :: RLVECT_5:31 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W")))) "iff" (Bool (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "W")))) ")" ))) ; theorem :: RLVECT_5:32 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool "(" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V")))) ")" )) ; theorem :: RLVECT_5:33 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool "(" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) ")" )) ")" )) ; theorem :: RLVECT_5:34 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool "(" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Num 2)) "iff" (Bool "ex" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set (Var "v"))) & (Bool (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "u")) "," (Set (Var "v")) ($#k7_domain_1 :::"}"::: ) ) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlvect_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "u")) "," (Set (Var "v")) ($#k7_domain_1 :::"}"::: ) ))) ")" )) ")" )) ; theorem :: RLVECT_5:35 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set "(" (Set (Var "W1")) ($#k1_rlsub_2 :::"+"::: ) (Set (Var "W2")) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set "(" (Set (Var "W1")) ($#k2_rlsub_2 :::"/\"::: ) (Set (Var "W2")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W2")) ")" ))))) ; theorem :: RLVECT_5:36 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set "(" (Set (Var "W1")) ($#k2_rlsub_2 :::"/\"::: ) (Set (Var "W2")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W2")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V")) ")" ))))) ; theorem :: RLVECT_5:37 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V")) ($#r1_rlsub_2 :::"is_the_direct_sum_of"::: ) (Set (Var "W1")) "," (Set (Var "W2")))) "holds" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W2")) ")" ))))) ; theorem :: RLVECT_5:38 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V")))) "iff" (Bool "ex" (Set (Var "W")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))) ")" ))) ; definitionlet "V" be ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::); let "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func "n" :::"Subspaces_of"::: "V" -> ($#m1_hidden :::"set"::: ) means :: RLVECT_5:def 3 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "W")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" "V" "st" (Bool "(" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) "n") ")" )) ")" )); end; :: deftheorem defines :::"Subspaces_of"::: RLVECT_5:def 3 : (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_rlvect_5 :::"Subspaces_of"::: ) (Set (Var "V")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "W")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" )) ")" )) ")" )))); theorem :: RLVECT_5:39 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V"))))) "holds" (Bool "not" (Bool (Set (Set (Var "n")) ($#k2_rlvect_5 :::"Subspaces_of"::: ) (Set (Var "V"))) "is" ($#v1_xboole_0 :::"empty"::: ) )))) ; theorem :: RLVECT_5:40 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "st" (Bool (Bool (Set ($#k1_rlvect_5 :::"dim"::: ) (Set (Var "V"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "n")) ($#k2_rlvect_5 :::"Subspaces_of"::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: RLVECT_5:41 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_5 :::"finite-dimensional"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "n")) ($#k2_rlvect_5 :::"Subspaces_of"::: ) (Set (Var "W"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "n")) ($#k2_rlvect_5 :::"Subspaces_of"::: ) (Set (Var "V"))))))) ;