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<b>let </b><font color="Maroon" title="c7">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <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="Maroon" title="c7">n</font></span>)</span><br/> for <font color="Olive" title="b3">r</font> being <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">r</font> holds <br/> <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b2">p</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> </span><br/><b>let </b><font color="Maroon" title="c8">o</font>, <font color="Maroon" title="c9">p</font> be   <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="Maroon" title="c7">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Olive" title="b1">r</font> holds <br/> <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Olive" title="b1">r</font>,<font color="Maroon" title="c9">p</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> </span><br/><b>let </b><font color="Maroon" title="c10">r</font> be  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font> implies  <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  )</span><br/>





<b>assume </b><a NAME="E1:99"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> </span><br/>

<b>set </b><font color="Maroon" title="c11">f</font> =  <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font>;<br/>
<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c12">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span><br/>

<b>assume </b><a NAME="E2:99"/><i><font color="Green" title="E74">A2</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/>

<b>set </b><font color="Maroon" title="c13">S</font> =  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>;<br/>
<a NAME="E3:99"/><i><font color="Green" title="E75">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">y</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="Maroon" title="c7">n</font></span>)</span><b> such that </b><br/><a NAME="E5:99"/><i><font color="Green" title="E76">A4</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font>
 <b>and </b><br/><a NAME="E6:99"/><i><font color="Green" title="E77">A5</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="#E1:99"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="#E2:99"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="jordan.html#D8" target="_self" title="JORDAN:def.8">Def8</a></i>;<br/>
<a NAME="E7:99"/><i><font color="Green" title="E78">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9" title="TOPREALB:th.9">TOPREALB:9</a></i>;<br/>
<a NAME="E8:99"/>
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17" title="TOPREAL9:th.17">TOPREAL9:17</a></i>;<br/>
<a NAME="E9:99"/><b>then </b><i><font color="Green" title="E79">A7</font></i>: 
<font color="Maroon" title="c14">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:99"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="#E3:99"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="#E5:99"><i><font color="Green" title="E76">A4</font></i></a>, <a class="txt" href="#E7:99"><i><font color="Green" title="E78">A6</font></i></a>, <a class="ref" href="brouwer.html#T3" title="BROUWER:th.3">BROUWER:3</a></i>;<br/>
<a NAME="E10:99"/>
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T16" title="TOPREAL9:th.16">TOPREAL9:16</a></i>;<br/>
<a NAME="E11:99"/><b>then </b><i><font color="Green" title="E80">A8</font></i>: 
<font color="Maroon" title="c9">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:99"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="brouwer.html#T3" title="BROUWER:th.3">BROUWER:3</a></i>;<br/>
<a NAME="E12:99"/>
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T19" title="TOPREAL9:th.19">TOPREAL9:19</a></i>;<br/>
<a NAME="E13:99"/><b>then </b>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p</font>
 
<b>by </b><i><a class="txt" href="#E1:99"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="#E2:99"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="#E5:99"><i><font color="Green" title="E76">A4</font></i></a>, <a class="txt" href="#E7:99"><i><font color="Green" title="E78">A6</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E14:99"/>
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="#E5:99"><i><font color="Green" title="E76">A4</font></i></a>, <a class="txt" href="#E6:99"><i><font color="Green" title="E77">A5</font></i></a>, <a class="txt" href="#E9:99"><i><font color="Green" title="E79">A7</font></i></a>, <a class="txt" href="#E11:99"><i><font color="Green" title="E80">A8</font></i></a>, <a class="ref" href="brouwer.html#D3" title="BROUWER:def.3">BROUWER:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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