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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E34">Th34</font></span>: <a NAME="T40"><span class="comment"><font color="firebrick">:: ANPROJ_8:40</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span><br/>  for <font color="Olive" title="b4">M</font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive" title="b4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a></span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">q</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">q</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">q</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a></span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">r</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">r</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">r</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a></span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp;  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b4">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 holds <br/> ex <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">c</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span> &amp; ( <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Olive" title="b6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Olive" title="b7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</div></div>
