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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E19">Th19</font></span>: <a NAME="T22"><span class="comment"><font color="firebrick">:: EUCLID13:22</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">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">An</font>, <font color="Olive" title="b4">Bn</font>, <font color="Olive" title="b5">Cn</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>  st <font color="Olive" title="b4">Bn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">Cn</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><font color="Olive" title="b4">Bn</font>,<font color="Olive" title="b5">Cn</font></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><font color="Olive" title="b5">Cn</font>,<font color="Olive" title="b5">Cn</font></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><font color="Olive" title="b3">An</font>,<font color="Olive" title="b4">Bn</font></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><font color="Olive" title="b3">An</font>,<font color="Olive" title="b5">Cn</font></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b4">Bn</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b5">Cn</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b4">Bn</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b5">Cn</font></span>)</span></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span></span>)</span> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><span class="p1"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">Bn</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b5">Cn</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">An</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b4">Bn</font></span>)</span></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </div></div>
