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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">r</font> be   non  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</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><br/>  for <font color="Olive" title="b2">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> holds  <font color="Olive" title="b2">f</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">o</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> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> holds  <font color="Olive" title="b1">f</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> </span><br/>





<a NAME="E1:23"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#T9" title="TOPREALB:th.9">TOPREALB:9</a></span>;<br/>
<a NAME="E2:23"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">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> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:23_1"/>
( <font color="Maroon" title="c2">r</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  or  not <font color="Maroon" title="c2">r</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1"/>
<font color="Maroon" title="c2">r</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c2">r</font> as   <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>  ;<br/><a NAME="E3:23_1_1"/>
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">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="c3">o</font>,<font color="Maroon" title="c5">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T17" title="TOPREAL9:th.17">TOPREAL9:17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Y</font> =  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">r</font>) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">r</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> =  <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">r</font>)</span>)</span>,<font color="Maroon" title="c6">Y</font> ;<br/><a NAME="E6:23_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 not <font color="Maroon" title="c6">Y</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a>  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="brouwer.html#T10" target="_self" title="BROUWER:th.10">Th10</a></span>;<br/><span class="kw">assume </span><a NAME="E7:23_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E8:23_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c7">g</font> is  <a href="borsuk_1.html#V3" title="BORSUK_1:attr.3">being_a_retraction</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">W</font> be   <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="c3">o</font>,<font color="Maroon" title="c5">r</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="borsuk_1.html#D16" title="BORSUK_1:def.16">BORSUK_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Y</font> or <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c8">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W</font></span><br/>

<a NAME="E2:23_1_1_1"/>
 not <font color="Maroon" title="c8">W</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="brouwer.html#T11" target="_self" title="BROUWER:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:23_1_1"/>
<font color="Maroon" title="c7">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="brouwer.html#T13" target="_self" title="BROUWER:th.13">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E10:23_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_2"/>
 not <font color="Maroon" title="c2">r</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c2">r</font> as   non  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   non  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</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>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">o</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">o</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c4">f</font></span><br/><a NAME="E3:23_1_2"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="brouwer.html#T2" target="_self" title="BROUWER:th.2">Th2</a></span>;<br/><a NAME="E4:23_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <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="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:23_1_2"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:23_1_2"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o</font></span><br/><a NAME="E6:23_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:23_1_2"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:23_1_2"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
