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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">A</font> being   <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</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> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  &amp;  not <font color="Olive" title="b1">A</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  holds <br/> for <font color="Olive" title="b2">h</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">A</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">A</font></span>)</span> holds  <font color="Olive" title="b2">h</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">T</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">I</font> being   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</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> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:13"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">I</font>
 <span class="kw">and </span><br/><a NAME="E3:13"/>
<font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:13"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T60" title="RLAFFIN1:th.60">RLAFFIN1:60</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">TT</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c3">I</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">TT</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">II</font> =  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c5">i</font>;<br/>
<a NAME="E7:13"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 not <font color="Maroon" title="c3">I</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E12">A1</font></span></a></span>;<br/>
<a NAME="E8:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 not  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c5">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T20" title="RLAFFIN2:th.20">RLAFFIN2:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">ii</font> =  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c5">i</font> as   <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> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E10:13"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">ii</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T5" title="RLAFFIN2:th.5">RLAFFIN2:5</a>, <a class="ref" href="tops_1.html#T19" title="TOPS_1:th.19">TOPS_1:19</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</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> <font color="Maroon" title="c1">n</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="c8">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  &amp;  not <font color="Maroon" title="c8">A</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  implies  for <font color="Olive" title="b1">h</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A</font></span>)</span> holds  <font color="Olive" title="b1">h</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E11:13"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
( <font color="Maroon" title="c8">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  &amp;  not <font color="Maroon" title="c8">A</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</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">h</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A</font></span>)</span> holds  <font color="Olive" title="b1">h</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> </span><br/>

<a NAME="E12:13"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
 not <font color="Maroon" title="c8">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:13"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">h</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A</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="c9">h</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> </span><br/>

<a NAME="E13:13"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_4.html#T22" title="RUSUB_4:th.22">RUSUB_4:22</a></span>;<br/>
<a NAME="E14:13"/><span class="kw">then </span>
(  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T50" title="RLAFFIN1:th.50">RLAFFIN1:50</a>, <a class="ref" href="rlaffin3.html#T4" title="RLAFFIN3:th.4">RLAFFIN3:4</a></span>;<br/>
<a NAME="E15:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="rlaffin3.html#T6" title="RLAFFIN3:th.6">RLAFFIN3:6</a></span>;<br/>
<a NAME="E16:13"/><span class="kw">then </span>
<font color="Maroon" title="c7">ii</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T33" title="RLAFFIN3:th.33">RLAFFIN3:33</a></span>;<br/>
<a NAME="E17:13"/><span class="kw">then </span>
 not  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E10:13"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="tops_1.html#T23" title="TOPS_1:th.23">TOPS_1:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E19:13"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c10">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E20:13"/>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:13"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="brouwer2.html#T7" target="_self" title="BROUWER2:th.7">Th7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">fhf</font> = <font color="Maroon" title="c10">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:13"><span class="lab"><font color="Green" title="E17">A6</font></span></a></span>;<br/>
<a NAME="E22:13"/>
<font color="Maroon" title="c10">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  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="#E19:13"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E24:13"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c12">p</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="c11">fhf</font>
 <span class="kw">and </span><br/><a NAME="E25:13"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c11">fhf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:13"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E7:13"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E19:13"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E12:13"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="simplex2.html#T23" title="SIMPLEX2:th.23">SIMPLEX2:23</a></span>;<br/>
<a NAME="E26:13"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c12">p</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:13"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">F</font> = <font color="Maroon" title="c10">f</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E28:13"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:13"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E29:13"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:13"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E31:13"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">F</font>
 <span class="kw">and </span><br/><a NAME="E32:13"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c13">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:13"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E24:13"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E33:13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:13"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E31:13"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="txt" href="#E32:13"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></span>;<br/>
<a NAME="E34:13"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:13"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E29:13"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E35:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A16</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:13"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E25:13"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E36:13"/><span class="lab"><font color="Green" title="E28">A17</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:13"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E37:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A18</font></span>: 
<font color="Maroon" title="c14">x</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="c9">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:13"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E38:13"/><span class="kw">then </span>
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</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="c9">h</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/>
<a NAME="E39:13"/><span class="kw">then </span>
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:13"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E36:13"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="txt" href="#E31:13"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="txt" href="#E32:13"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E35:13"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E40:13"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c9">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:13"><span class="lab"><font color="Green" title="E29">A18</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E41:13"/>
<font color="Maroon" title="c9">h</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</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>
