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<div><span class="kw">theorem </span><a NAME="T59"><span class="comment"><font color="firebrick">:: BORSUK_4:59</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">D</font> being   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a><br/>  for <font color="Olive" title="b2">C</font> being   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b1">D</font> holds <br/> ex <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</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> 2</span>)</span> ex <font color="Olive" title="b5">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b5">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font> &amp; <font color="Olive" title="b2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">B</font> &amp; <font color="Olive" title="b5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> )</div></div>
