:: VECTSP_7  semantic presentation begin  





































definitionlet "GF" be   ($#l6_algstr_0 :::"Field":::); let "V" be   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Const "GF")); let "IT" be   ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "IT" is  :::"linearly-independent":::  means :: VECTSP_7:def 1 (Bool  "for" (Set (Var "l")) "being"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" "IT"  "st" (Bool (Bool (Set   ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")))  ($#r1_hidden :::"="::: )  (Set   ($#k4_struct_0 :::"0."::: )  "V"))) "holds"  (Bool (Set   ($#k1_vectsp_6 :::"Carrier"::: )  (Set (Var "l")))  ($#r1_hidden :::"="::: )  (Set   ($#k1_xboole_0 :::"{}"::: )  ))); end; 






:: deftheorem    defines :::"linearly-independent"::: VECTSP_7:def 1 :  (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "IT")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set (Var "IT")) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  ) "iff" (Bool  "for" (Set (Var "l")) "being"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" (Set (Var "IT"))  "st" (Bool (Bool (Set   ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")))  ($#r1_hidden :::"="::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V"))))) "holds"  (Bool (Set   ($#k1_vectsp_6 :::"Carrier"::: )  (Set (Var "l")))  ($#r1_hidden :::"="::: )  (Set   ($#k1_xboole_0 :::"{}"::: )  )))  ")" )))); 
 notationlet "GF" be   ($#l6_algstr_0 :::"Field":::); let "V" be   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Const "GF")); let "IT" be   ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); antonym  :::"linearly-dependent"::: "IT" for  :::"linearly-independent"::: ; end; 
theorem :: VECTSP_7:1 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (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"))) & (Bool (Set (Var "B")) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  )) "holds"  (Bool (Set (Var "A")) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  )))) ; 
 
theorem :: VECTSP_7:2 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set (Var "A")) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  )) "holds"  (Bool  "not" (Bool (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V")))  ($#r2_hidden :::"in"::: )  (Set (Var "A"))))))) ; 
 registrationlet "GF" be   ($#l6_algstr_0 :::"Field":::); let "V" be   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Const "GF")); 
cluster   ($#v1_xboole_0 :::"empty"::: )    ->   ($#v1_vectsp_7 :::"linearly-independent"::: )    for    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k1_zfmisc_1 :::"bool"::: )  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" "V")); 
end; registrationlet "GF" be   ($#l6_algstr_0 :::"Field":::); let "V" be   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Const "GF")); 
cluster   ($#v1_finset_1 :::"finite"::: )     ($#v1_vectsp_7 :::"linearly-independent"::: )    for    ($#m1_subset_1 :::"Element"::: )   "of" (Set   ($#k1_zfmisc_1 :::"bool"::: )  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" "V")); 
end; 
theorem :: VECTSP_7:3 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "v")) "being"   ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set  ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  ) "iff" (Bool (Set (Var "v"))  ($#r1_hidden :::"<>"::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V"))))  ")" )))) ; 
 
theorem :: VECTSP_7:4 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "v1")) "," (Set (Var "v2")) "being"   ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set  ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  )) "holds"  (Bool (Set (Var "v1"))  ($#r1_hidden :::"<>"::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V"))))))) ; 
 
theorem :: VECTSP_7:5 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "v1")) "," (Set (Var "v2")) "being"   ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set (Var "v1"))  ($#r1_hidden :::"<>"::: )  (Set (Var "v2"))) & (Bool (Set  ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  ) "iff" (Bool  "("  (Bool (Set (Var "v2"))  ($#r1_hidden :::"<>"::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V")))) & (Bool  "("   "for" (Set (Var "a")) "being"   ($#m1_subset_1 :::"Element":::) "of" (Set (Var "GF"))  "holds" (Bool (Set (Var "v1"))  ($#r1_hidden :::"<>"::: )  (Set (Set (Var "a"))  ($#k4_vectsp_1 :::"*"::: )  (Set (Var "v2"))))  ")" )  ")" )  ")" )))) ; 
 
theorem :: VECTSP_7:6 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "v1")) "," (Set (Var "v2")) "being"   ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool  "("  (Bool (Set (Var "v1"))  ($#r1_hidden :::"<>"::: )  (Set (Var "v2"))) & (Bool (Set  ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  )  ")" ) "iff" (Bool  "for" (Set (Var "a")) "," (Set (Var "b")) "being"   ($#m1_subset_1 :::"Element":::) "of" (Set (Var "GF"))  "st" (Bool (Bool (Set (Set  "(" (Set (Var "a"))  ($#k4_vectsp_1 :::"*"::: )  (Set (Var "v1")) ")" )  ($#k3_rlvect_1 :::"+"::: )  (Set  "(" (Set (Var "b"))  ($#k4_vectsp_1 :::"*"::: )  (Set (Var "v2")) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V"))))) "holds"  (Bool  "("  (Bool (Set (Var "a"))  ($#r1_hidden :::"="::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "GF")))) & (Bool (Set (Var "b"))  ($#r1_hidden :::"="::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "GF"))))  ")" ))  ")" )))) ; 
 definitionlet "GF" be   ($#l6_algstr_0 :::"Field":::); let "V" be   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Const "GF")); let "A" be   ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func  :::"Lin"::: "A" ->   ($#v7_vectsp_1 :::"strict"::: )     ($#m1_vectsp_4 :::"Subspace"::: )   "of" "V" means :: VECTSP_7:def 2 (Bool (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" it)  ($#r1_hidden :::"="::: )   "{"  (Set  "("  ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")) ")" ) where l "is"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" "A" : (Bool verum)  "}"  ); end; 







:: deftheorem    defines :::"Lin"::: VECTSP_7:def 2 :  (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  (Bool  "for" (Set (Var "b4")) "being"   ($#v7_vectsp_1 :::"strict"::: )     ($#m1_vectsp_4 :::"Subspace"::: )   "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set (Var "b4"))  ($#r1_hidden :::"="::: )  (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")))) "iff" (Bool (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "b4")))  ($#r1_hidden :::"="::: )   "{"  (Set  "("  ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")) ")" ) where l "is"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" (Set (Var "A")) : (Bool verum)  "}"  )  ")" ))))); 
 
theorem :: VECTSP_7:7 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set (Var "x"))  ($#r1_struct_0 :::"in"::: )  (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")))) "iff" (Bool  "ex" (Set (Var "l")) "being"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" (Set (Var "A")) "st" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Set   ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")))))  ")" ))))) ; 
 
theorem :: VECTSP_7:8 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set (Var "A")))) "holds"  (Bool (Set (Var "x"))  ($#r1_struct_0 :::"in"::: )  (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")))))))) ; 
 



theorem :: VECTSP_7:9 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF")) "holds"  (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set  "("  ($#k1_subset_1 :::"{}"::: )  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "V"))) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k1_vectsp_4 :::"(0)."::: )  (Set (Var "V")))))) ; 
 
theorem :: VECTSP_7:10 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "holds"  (Bool  "("   "not" (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set   ($#k1_vectsp_4 :::"(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 :: VECTSP_7:11 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  (Bool  "for" (Set (Var "W")) "being"   ($#v7_vectsp_1 :::"strict"::: )     ($#m1_vectsp_4 :::"Subspace"::: )   "of" (Set (Var "V"))  "st" (Bool (Bool (Set (Var "A"))  ($#r1_hidden :::"="::: )  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "W"))))) "holds"  (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set (Var "W"))))))) ; 
 
theorem :: VECTSP_7:12 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#v7_vectsp_1 :::"strict"::: )    ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#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   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set (Var "V")))))) ; 
 
theorem :: VECTSP_7:13 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (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   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A"))) "is"    ($#m1_vectsp_4 :::"Subspace"::: )   "of" (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "B"))))))) ; 
 
theorem :: VECTSP_7:14 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#v7_vectsp_1 :::"strict"::: )    ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "," (Set (Var "B")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set (Var "V"))) & (Bool (Set (Var "A"))  ($#r1_tarski :::"c="::: )  (Set (Var "B")))) "holds"  (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "B")))  ($#r1_hidden :::"="::: )  (Set (Var "V")))))) ; 
 
theorem :: VECTSP_7:15 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "," (Set (Var "B")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"  (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set  "(" (Set (Var "A"))  ($#k4_subset_1 :::"\/"::: )  (Set (Var "B")) ")" ))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")) ")" )  ($#k1_vectsp_5 :::"+"::: )  (Set  "("  ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "B")) ")" )))))) ; 
 
theorem :: VECTSP_7:16 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "," (Set (Var "B")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"  (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set  "(" (Set (Var "A"))  ($#k9_subset_1 :::"/\"::: )  (Set (Var "B")) ")" )) "is"    ($#m1_vectsp_4 :::"Subspace"::: )   "of" (Set (Set  "("  ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")) ")" )  ($#k2_vectsp_5 :::"/\"::: )  (Set  "("  ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "B")) ")" )))))) ; 
 
theorem :: VECTSP_7:17 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set (Var "A")) "is"   ($#v1_vectsp_7 :::"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"   ($#v1_vectsp_7 :::"linearly-independent"::: )  ) & (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "B")))  ($#r1_hidden :::"="::: )  (Set   ($#g1_vectsp_1 :::"VectSpStr"::: )  "(#"  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "V"))) "," (Set  "the"  ($#u1_algstr_0 :::"U5"::: )  "of" (Set (Var "V"))) "," (Set  "the"  ($#u2_struct_0 :::"ZeroF"::: )  "of" (Set (Var "V"))) "," (Set  "the"  ($#u1_vectsp_1 :::"lmult"::: )  "of" (Set (Var "V"))) "#)" ))  ")" ))))) ; 
 
theorem :: VECTSP_7:18 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k1_vectsp_7 :::"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_vectsp_7 :::"linearly-independent"::: )  ) & (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "B")))  ($#r1_hidden :::"="::: )  (Set (Var "V")))  ")" ))))) ; 
 definitionlet "GF" be   ($#l6_algstr_0 :::"Field":::); let "V" be   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Const "GF")); mode  :::"Basis":::  "of" "V" ->   ($#m1_subset_1 :::"Subset":::) "of" "V" means :: VECTSP_7:def 3 (Bool  "("  (Bool it "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  ) & (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  it)  ($#r1_hidden :::"="::: )  (Set   ($#g1_vectsp_1 :::"VectSpStr"::: )  "(#"  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" "V") "," (Set  "the"  ($#u1_algstr_0 :::"U5"::: )  "of" "V") "," (Set  "the"  ($#u2_struct_0 :::"ZeroF"::: )  "of" "V") "," (Set  "the"  ($#u1_vectsp_1 :::"lmult"::: )  "of" "V") "#)" ))  ")" ); end; 






:: deftheorem    defines :::"Basis"::: VECTSP_7:def 3 :  (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "b3")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set (Var "b3")) "is"    ($#m1_vectsp_7 :::"Basis"::: )   "of" (Set (Var "V"))) "iff" (Bool  "("  (Bool (Set (Var "b3")) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  ) & (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "b3")))  ($#r1_hidden :::"="::: )  (Set   ($#g1_vectsp_1 :::"VectSpStr"::: )  "(#"  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "V"))) "," (Set  "the"  ($#u1_algstr_0 :::"U5"::: )  "of" (Set (Var "V"))) "," (Set  "the"  ($#u2_struct_0 :::"ZeroF"::: )  "of" (Set (Var "V"))) "," (Set  "the"  ($#u1_vectsp_1 :::"lmult"::: )  "of" (Set (Var "V"))) "#)" ))  ")" )  ")" )))); 
 
theorem :: VECTSP_7:19 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set (Var "A")) "is"   ($#v1_vectsp_7 :::"linearly-independent"::: )  )) "holds"  (Bool  "ex" (Set (Var "I")) "being"    ($#m1_vectsp_7 :::"Basis"::: )   "of" (Set (Var "V")) "st" (Bool (Set (Var "A"))  ($#r1_tarski :::"c="::: )  (Set (Var "I"))))))) ; 
 
theorem :: VECTSP_7:20 (Bool  "for" (Set (Var "GF")) "being"   ($#l6_algstr_0 :::"Field":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "GF"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k1_vectsp_7 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set (Var "V")))) "holds"  (Bool  "ex" (Set (Var "I")) "being"    ($#m1_vectsp_7 :::"Basis"::: )   "of" (Set (Var "V")) "st" (Bool (Set (Var "I"))  ($#r1_tarski :::"c="::: )  (Set (Var "A"))))))) ;