<?xml version="1.0" encoding="UTF-8"?>
<?xml-stylesheet type="text/xsl" href="../../../../xsl/longlemmainfo.xsl"?>
<LONGLEMMAINFO name="correct"><LEMMAINFO state="proved"><LEMMANAME>correct</LEMMANAME><LEMMAGOAL><SEQGOAL><GOALSEQ><SEQ>okspmx(spmx, r, &#35; x) ⊦ ⟪MULT&#35;(spmx, r, x; ; y)⟫ (y = spmx.toMatrix(r, &#35; x) * x)</SEQ></GOALSEQ></SEQGOAL></LEMMAGOAL><LEMMATYPE><OBLIGATIONLEMMA/></LEMMATYPE><USEDLEMMAS><LIST><LE>multzero</LE><LE>okspmx-def</LE><LE>main-lemma</LE><LE>toMatrix-rowno</LE><LE>toMatrix-colno</LE><LE>addzero</LE><LE>MULT</LE></LIST></USEDLEMMAS><SMTINFOS><LIST></LIST></SMTINFOS><MAINGOALS><LIST></LIST></MAINGOALS><USERACTIONS>8</USERACTIONS><PROOFSTEPS>30</PROOFSTEPS><PROVEDP><T/></PROVEDP><PROOFEXISTSP><T/></PROOFEXISTSP><PROOFFILENAME>correct-proof</PROOFFILENAME><LEMMAPROOFBAG><NONE/></LEMMAPROOFBAG><SAVETREEP><F/></SAVETREEP><INFOFILENAME>correct-proof-info</INFOFILENAME><LEMMAPROOFINFOBAG><NONE/></LEMMAPROOFINFOBAG><SAVEINFOSP><F/></SAVEINFOSP><SIMPFEATURES><LIST></LIST></SIMPFEATURES><HISTINFO><LIST></LIST></HISTINFO></LEMMAINFO><VALIDITY>The proof is valid.</VALIDITY><FEATURES><LIST></LIST></FEATURES><HISTORY><LIST></LIST></HISTORY><USEDBY><LIST></LIST></USEDBY><USEDBYUNITS><LIST></LIST></USEDBYUNITS><COMMENT></COMMENT><USED_SPECLEMS><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/matrixops/export/unit.xml</HTMLTARGET><HTMLSTRING>matrixops</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/matrixops/export/multmxvec-size/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>multmxvec-size</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/Extension/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>Extension</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/matrixops/export/unit.xml</HTMLTARGET><HTMLSTRING>matrixops</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/matrixops/export/addvec-com/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>addvec-com</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ></USED_SPECLEMS><USED_SIMPRULES><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/array/export/unit.xml</HTMLTARGET><HTMLSTRING>array</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>matrixops</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/array/export/Size-mkarray/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>Size-mkarray</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/sub-ref/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>sub-ref</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/f/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>f</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/c/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>c</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/a/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>a</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/ref/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>ref</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/s-04/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>s-04</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/ls-01/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>ls-01</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/empty/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>empty</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/one-01/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>one-01</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/le-def/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>le-def</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/pred-03/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>pred-03</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/size/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>size</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/array/export/unit.xml</HTMLTARGET><HTMLSTRING>array</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>matrixops</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/array/export/Size-put/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>Size-put</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/subset-delete-02/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>subset-delete-02</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/ring/export/unit.xml</HTMLTARGET><HTMLSTRING>ring</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/ring/export/multassoc/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>multassoc</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/ring/export/unit.xml</HTMLTARGET><HTMLSTRING>ring</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/ring/export/addassoc/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>addassoc</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/ring/export/unit.xml</HTMLTARGET><HTMLSTRING>ring</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/ring/export/addcom/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>addcom</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../specs/natpair/export/unit.xml</HTMLTARGET><HTMLSTRING>natpair</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../specs/natpair/export/elim/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>elim</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/lels/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>lels</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../specs/natpair/export/unit.xml</HTMLTARGET><HTMLSTRING>natpair</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../specs/natpair/export/sel-01/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>sel-01</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../specs/natpair/export/unit.xml</HTMLTARGET><HTMLSTRING>natpair</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../specs/natpair/export/sel-02/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>sel-02</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/le-ls-simp/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>le-ls-simp</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/superset-elem-in-01/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>superset-elem-in-01</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/notmemlookup/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>notmemlookup</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/In-empty/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>In-empty</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/unit.xml</HTMLTARGET><HTMLSTRING>store</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>sparsearray</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/store/export/subset-empty/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>subset-empty</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ></USED_SIMPRULES></LONGLEMMAINFO>