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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E54">Th48</font></span>: <a NAME="T57"><span class="comment"><font color="firebrick">:: BKMODEL4:57</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   non  <a href="bkmodel4.html#V1" title="BKMODEL4:attr.1">point_at_infty</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span></span>)</span><br/>  for <font color="Olive" title="b4">u</font>, <font color="Olive" title="b5">v</font>, <font color="Olive" title="b6">w</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span>  st <font color="Olive" title="b1">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bkmodel1.html#K7" title="BKMODEL1:func.7">absolute</a>  &amp; <font color="Olive" title="b2">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bkmodel1.html#K7" title="BKMODEL1:func.7">absolute</a>  &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b4">u</font> &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b5">v</font> &amp; <font color="Olive" title="b3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b6">w</font> &amp; <font color="Olive" title="b4">u</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b5">v</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b4">u</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b5">v</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b4">u</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b5">v</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> holds <br/><font color="Olive" title="b3">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bkmodel2.html#K1" title="BKMODEL2:func.1">BK_model</a> </div></div>
