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<b>let </b><font color="Maroon">r</font> be  non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">o</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">x</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E3:16"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="abian.html#D4">ABIAN:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E5:16"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp;  <a href="brouwer.html#K4" target="_self">HC</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>,<font color="Maroon">o</font>,<font color="Maroon">r</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:16"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K4">halfline</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="topreal9.html#T28">TOPREAL9:28</a></i>;<br/>
<a NAME="E7:16"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9">TOPREALB:9</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4">halfline</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:16"/>
 <a href="brouwer.html#K4" target="_self">HC</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


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