:: RUSUB_3 semantic presentation begin definitionlet "V" be ($#l1_bhsp_1 :::"RealUnitarySpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"Lin"::: "A" -> ($#v1_bhsp_1 :::"strict"::: ) ($#m1_rusub_1 :::"Subspace"::: ) "of" "V" means :: RUSUB_3:def 1 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "l")) ")" ) where l "is" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" "A" : (Bool verum) "}" ); end; :: deftheorem defines :::"Lin"::: RUSUB_3:def 1 : (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "b3")) "being" ($#v1_bhsp_1 :::"strict"::: ) ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A")))) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "l")) ")" ) where l "is" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "A")) : (Bool verum) "}" ) ")" )))); theorem :: RUSUB_3:1 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A")))) "iff" (Bool "ex" (Set (Var "l")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "l"))))) ")" )))) ; theorem :: RUSUB_3:2 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))))))) ; theorem :: RUSUB_3:3 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) "holds" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set "(" ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_rusub_1 :::"(0)."::: ) (Set (Var "V"))))) ; theorem :: RUSUB_3:4 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" "not" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rusub_1 :::"(0)."::: ) (Set (Var "V")))) "or" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ")" ))) ; theorem :: RUSUB_3:5 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#v1_bhsp_1 :::"strict"::: ) ($#m1_rusub_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_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))) "holds" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "W")))))) ; theorem :: RUSUB_3:6 (Bool "for" (Set (Var "V")) "being" ($#v1_bhsp_1 :::"strict"::: ) ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))))) "holds" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "V"))))) ; theorem :: RUSUB_3:7 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "B")))))) ; theorem :: RUSUB_3:8 (Bool "for" (Set (Var "V")) "being" ($#v1_bhsp_1 :::"strict"::: ) ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "V"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set (Var "V"))))) ; theorem :: RUSUB_3:9 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A")) ")" ) ($#k1_rusub_2 :::"+"::: ) (Set "(" ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "B")) ")" ))))) ; theorem :: RUSUB_3:10 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" )) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Set "(" ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A")) ")" ) ($#k2_rusub_2 :::"/\"::: ) (Set "(" ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "B")) ")" ))))) ; theorem :: RUSUB_3:11 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) )) "holds" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Var "B")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#g1_bhsp_1 :::"UNITSTR"::: ) "(#" (Set "the" ($#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" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_bhsp_1 :::"scalar"::: ) "of" (Set (Var "V"))) "#)" )) ")" )))) ; theorem :: RUSUB_3:12 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "B")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set (Var "V"))) ")" )))) ; begin definitionlet "V" be ($#l1_bhsp_1 :::"RealUnitarySpace":::); mode :::"Basis"::: "of" "V" -> ($#m1_subset_1 :::"Subset":::) "of" "V" means :: RUSUB_3:def 2 (Bool "(" (Bool it "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#g1_bhsp_1 :::"UNITSTR"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" "V") "," (Set "the" ($#u1_algstr_0 :::"U5"::: ) "of" "V") "," (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" "V") "," (Set "the" ($#u1_bhsp_1 :::"scalar"::: ) "of" "V") "#)" )) ")" ); end; :: deftheorem defines :::"Basis"::: RUSUB_3:def 2 : (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_rusub_3 :::"Basis"::: ) "of" (Set (Var "V"))) "iff" (Bool "(" (Bool (Set (Var "b2")) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#g1_bhsp_1 :::"UNITSTR"::: ) "(#" (Set "the" ($#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" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_bhsp_1 :::"scalar"::: ) "of" (Set (Var "V"))) "#)" )) ")" ) ")" ))); theorem :: RUSUB_3:13 (Bool "for" (Set (Var "V")) "being" ($#v1_bhsp_1 :::"strict"::: ) ($#l1_bhsp_1 :::"RealUnitarySpace":::) (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" ($#m1_rusub_3 :::"Basis"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "I")))))) ; theorem :: RUSUB_3:14 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "I")) "being" ($#m1_rusub_3 :::"Basis"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))))) ; theorem :: RUSUB_3:15 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#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_rusub_3 :::"Basis"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "I")))))) ; begin theorem :: RUSUB_3:16 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (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" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A")) ")" )))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#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 :: RUSUB_3:17 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "L")) "being" ($#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 "(" ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A")) ")" )))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "A")) "st" (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "K")))))))) ; theorem :: RUSUB_3:18 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#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 ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "K")))) ")" ))))) ; theorem :: RUSUB_3:19 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "K")) "being" ($#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 ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L")))) ")" ))))) ; theorem :: RUSUB_3:20 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#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 :: RUSUB_3:21 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "I")) "being" ($#m1_rusub_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 ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "I"))))))) ; theorem :: RUSUB_3:22 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#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"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")))))) ; theorem :: RUSUB_3:23 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#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"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "W")))))) ; theorem :: RUSUB_3:24 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_rusub_3 :::"Basis"::: ) "of" (Set (Var "W")) (Bool "ex" (Set (Var "B")) "being" ($#m1_rusub_3 :::"Basis"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))))))) ; theorem :: RUSUB_3:25 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "A")) "being" ($#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_rusub_3 :::"Lin"::: ) (Set (Var "B"))))))))) ; theorem :: RUSUB_3:26 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "I")) "being" ($#m1_rusub_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 :: RUSUB_3:27 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rusub_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_rusub_3 :::"Lin"::: ) (Set (Var "A"))) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W")))))) ; theorem :: RUSUB_3:28 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#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 ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set (Var "B")))))))) ; begin theorem :: RUSUB_3:29 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v"))))) ")" )))) ; theorem :: RUSUB_3:30 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: RUSUB_3:31 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rusub_1 :::"+"::: ) (Set "(" ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "w")) ($#k6_domain_1 :::"}"::: ) ) ")" ))) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w")) ")" )))) ")" )))) ; theorem :: RUSUB_3:32 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) ($#k7_domain_1 :::"}"::: ) ))) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w2")) ")" )))) ")" )))) ; theorem :: RUSUB_3:33 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "w1")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) ($#k7_domain_1 :::"}"::: ) ))) & (Bool (Set (Var "w2")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) ($#k7_domain_1 :::"}"::: ) ))) ")" ))) ; theorem :: RUSUB_3:34 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rusub_1 :::"+"::: ) (Set "(" ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) ($#k7_domain_1 :::"}"::: ) ) ")" ))) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w1")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w2")) ")" )))) ")" )))) ; theorem :: RUSUB_3:35 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) ($#k8_domain_1 :::"}"::: ) ))) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "c")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v3")) ")" )))) ")" )))) ; theorem :: RUSUB_3:36 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "," (Set (Var "w3")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "w1")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) "," (Set (Var "w3")) ($#k8_domain_1 :::"}"::: ) ))) & (Bool (Set (Var "w2")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) "," (Set (Var "w3")) ($#k8_domain_1 :::"}"::: ) ))) & (Bool (Set (Var "w3")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) "," (Set (Var "w3")) ($#k8_domain_1 :::"}"::: ) ))) ")" ))) ; theorem :: RUSUB_3:37 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w1")) "," (Set (Var "w2")) "," (Set (Var "w3")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rusub_1 :::"+"::: ) (Set "(" ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) "," (Set (Var "w3")) ($#k8_domain_1 :::"}"::: ) ) ")" ))) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w1")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w2")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "c")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w3")) ")" )))) ")" )))) ; theorem :: RUSUB_3:38 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "w")) ($#k6_domain_1 :::"}"::: ) ))) & (Bool (Set (Var "v")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))) "holds" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "w")) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: RUSUB_3:39 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2"))) & (Bool (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) ($#k7_domain_1 :::"}"::: ) ))) & (Bool (Set (Var "v2")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) ($#k7_domain_1 :::"}"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) ($#k7_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rusub_3 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ))) & (Bool (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "w1")) "," (Set (Var "w2")) ($#k7_domain_1 :::"}"::: ) ) "is" ($#v1_rlvect_3 :::"linearly-independent"::: ) ) & (Bool (Set (Var "w1")) ($#r1_hidden :::"<>"::: ) (Set (Var "w2"))) ")" ))) ; begin theorem :: RUSUB_3:40 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rusub_1 :::"(0)."::: ) (Set (Var "V")))) "iff" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) ")" ))) ; theorem :: RUSUB_3:41 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "," (Set (Var "W3")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "W1")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W3")))) "holds" (Bool (Set (Set (Var "W1")) ($#k2_rusub_2 :::"/\"::: ) (Set (Var "W2"))) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W3"))))) ; theorem :: RUSUB_3:42 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "," (Set (Var "W3")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "W1")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W2"))) & (Bool (Set (Var "W1")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W3")))) "holds" (Bool (Set (Var "W1")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Set (Var "W2")) ($#k2_rusub_2 :::"/\"::: ) (Set (Var "W3")))))) ; theorem :: RUSUB_3:43 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "," (Set (Var "W3")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "W1")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W3"))) & (Bool (Set (Var "W2")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W3")))) "holds" (Bool (Set (Set (Var "W1")) ($#k1_rusub_2 :::"+"::: ) (Set (Var "W2"))) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W3"))))) ; theorem :: RUSUB_3:44 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "," (Set (Var "W3")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "W1")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W2")))) "holds" (Bool (Set (Var "W1")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Set (Var "W2")) ($#k1_rusub_2 :::"+"::: ) (Set (Var "W3")))))) ; theorem :: RUSUB_3:45 (Bool "for" (Set (Var "V")) "being" ($#l1_bhsp_1 :::"RealUnitarySpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "W1")) "is" ($#m1_rusub_1 :::"Subspace"::: ) "of" (Set (Var "W2")))) "holds" (Bool (Set (Set (Var "v")) ($#k3_rusub_1 :::"+"::: ) (Set (Var "W1"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "v")) ($#k3_rusub_1 :::"+"::: ) (Set (Var "W2"))))))) ;