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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E18">Th16</font></span>: <a NAME="T17"><span class="comment"><font color="firebrick">:: BROUWER3:17</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">p</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> <font color="Olive" title="b1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b4">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>( <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> (<font color="Olive" title="b4">s</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">s</font></span>)</span>) &amp; <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> (<font color="Olive" title="b4">s</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">s</font></span>)</span>) &amp; <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> (<font color="Olive" title="b4">s</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">s</font></span>)</span>) )</div></div>
