<?xml version="1.0"?>
<div class="add">

<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="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>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</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 holds <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"><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="Olive" title="b1">A</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="Olive" title="b1">A</font></span>)</span></span>)</span>  ex <font color="Olive" title="b3">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> st <br/>( <font color="Olive" title="b3">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="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">TRn</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</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 implies  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"><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">A</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">A</font></span>)</span></span>)</span>  ex <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="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b2">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="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</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
 ; <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"><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">A</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">A</font></span>)</span></span>)</span>  ex <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="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b2">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="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> )</span><br/>

<a NAME="E2:42"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
 not <font color="Maroon" title="c3">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="#E1:42"><span class="lab"><font color="Green" title="E28">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">cA</font> =  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">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"><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">A</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">A</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">cA</font> =  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">T</font> = <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="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">cA</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">ff</font> = <font color="Maroon" title="c5">f</font> as   <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="c6">cA</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="c6">cA</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">E</font> =  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c6">cA</font></span>)</span> : <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> </span> } </span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">F</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E6:42"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<a NAME="E7:42"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c6">cA</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font> or <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c6">cA</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</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="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c6">cA</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F</font>
 <span class="kw">and </span><br/><a NAME="E4:42_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c10">F</font> <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> <font color="Maroon" title="c11">y</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="E5:42_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c6">cA</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>) or <font color="Maroon" title="c13">z</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 <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="c6">cA</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</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="c13">z</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 <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="c6">cA</font></span>)</span></span><br/>

<a NAME="E2:42_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c6">cA</font></span>)</span> st <br/>( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:42_1_1"/>
<font color="Maroon" title="c13">z</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 <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="c6">cA</font></span>)</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="E6:42_1"/>
<font color="Maroon" title="c10">F</font> <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> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:42_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c6">cA</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E33">A6</font></span></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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c10">F</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c6">cA</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E9:42"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
 <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="c6">cA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">cA</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E10:42"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">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"><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="c6">cA</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c6">cA</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="c12">W</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="c11">F</font> implies <font color="Maroon" title="c12">W</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">Z</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  as   <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">reconsider </span><font color="Maroon" title="c14">L</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>,<font color="Maroon" title="c13">Z</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><span class="kw">assume </span><a NAME="E3:42_2"/>
<font color="Maroon" title="c12">W</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="c11">F</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="c12">W</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:42_2"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F</font>
 <span class="kw">and </span><br/><a NAME="E6:42_2"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W</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">reconsider </span><font color="Maroon" title="c16">AEi</font> =  <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span>) as   <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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="rlaffin3.html#T32" title="RLAFFIN3:th.32">RLAFFIN3:32</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">AT</font> = <font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1: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="c6">cA</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">Af</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">AfT</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span>;<br/><a NAME="E8:42_2"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p4">(<span class="default"><span class="p5">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span></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"><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="c6">cA</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">reconsider </span><font color="Maroon" title="c20">AfT</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></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> <font color="Maroon" title="c6">cA</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><a NAME="E10:42_2"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></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"><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="c6">cA</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/><a NAME="E11:42_2"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1: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="c6">cA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">AEi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c6">cA</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="tmap_1.html#D3" title="TMAP_1:def.3">TMAP_1:def 3</a></span>;<br/><a NAME="E12:42_2"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</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"><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="c6">cA</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/><a NAME="E13:42_2"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c20">AfT</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">i</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">AfT</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">L</font> or <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">i</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_2_1"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">AfT</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">L</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="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">i</font>)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">v</font> = <font color="Maroon" title="c21">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c6">cA</font></span>)</span> ;<br/>
<a NAME="E3:42_2_1"/>
<font color="Maroon" title="c21">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="c20">AfT</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E4:42_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<font color="Maroon" title="c20">AfT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p4">(<span class="default"><span class="p5">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>;<br/>
<a NAME="E5:42_2_1"/>
( <font color="Maroon" title="c20">AfT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">L</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_2"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E1:42_2_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E6:42_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/>
<a NAME="E7:42_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E8:42_2_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">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> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_2"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E1:42_2_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E9:42_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">AEi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42_2"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E10:42_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">AEi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_2"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E11:42_2"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:42_2_1"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E9:42_2_1"><span class="lab"><font color="Green" title="E43">A19</font></span></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="E14:42_2"/><span class="lab"><font color="Green" title="E40">A20</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">i</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">AfT</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">i</font>) or <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">AfT</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_2_2"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">i</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="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">AfT</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">L</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c6">cA</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:42_2_2"/><span class="lab"><font color="Green" title="E41">A21</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">and </span><br/><a NAME="E4:42_2_2"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
<font color="Maroon" title="c16">AEi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">AEi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 ;<br/>
<a NAME="E5:42_2_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</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="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E32">A8</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="E6:42_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A23</font></span>: 
<font color="Maroon" title="c16">AEi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_2"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E11:42_2"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E7:42_2_2"/>
<font color="Maroon" title="c16">AEi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_2"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E11:42_2"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E8:42_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_2"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E4:42_2_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E6:42_2_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E9:42_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p4">(<span class="default"><span class="p5">(<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="c6">cA</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">ff</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">v</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">AEi</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">cA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T47" title="XREAL_1:th.47">XREAL_1:47</a></span>;<br/>
<a NAME="E10:42_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">AfT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>;<br/>
<a NAME="E11:42_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">AfT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:42_2_2"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">AfT</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E3:42_2_2"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</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="E15:42_2"/>
<font color="Maroon" title="c14">L</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T41" title="BORSUK_5:th.41">BORSUK_5:41</a></span>;<br/><a NAME="E16:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A24</font></span>: 
<font color="Maroon" title="c20">AfT</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">L</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></span>;<br/><a NAME="E17:42_2"/>
<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E5:42_2"><span class="lab"><font color="Green" title="E33">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:42_2"/>
<font color="Maroon" title="c12">W</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_2"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E13:42_2"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E14:42_2"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E16:42_2"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="E12:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A25</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">F</font> is  <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a></span>;<br/>
<a NAME="E13:42"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E14:42"/><span class="lab"><font color="Green" title="E35">A27</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T65" title="RLAFFIN1:th.65">RLAFFIN1:65</a></span>;<br/>
<a NAME="E15:42"/><span class="lab"><font color="Green" title="E36">A28</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" title="RLAFFIN3:def.1">RLAFFIN3:def 1</a></span>;<br/>
<a NAME="E16:42"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<a NAME="E17:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A29</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E18:42"/>
 for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F</font></span>)</span> holds   <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">S</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">ES</font> =  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_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="E1:42_3_1"/>
( <font color="Maroon" title="c12">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c12">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_3_1_1"/>
<font color="Maroon" title="c12">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_3_1_1"/><span class="kw">then </span>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:42_3_1_1"/>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</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="42_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_3_1_2"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
 not <font color="Maroon" title="c12">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> )</span><br/><span class="kw">set </span><font color="Maroon" title="c15">fx</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>;<br/><span class="kw">set </span><font color="Maroon" title="c16">xES</font> = <font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">fxA</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font>;<br/><span class="kw">set </span><font color="Maroon" title="c18">xA</font> = <font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font>;<br/><span class="kw">assume </span><a NAME="E2:42_3_1_2"/><span class="lab"><font color="Green" title="E39">A31</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span><br/><a NAME="E3:42_3_1_2"/><span class="lab"><font color="Green" title="E40">A32</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a>, <a class="ref" href="rlaffin1.html#T3" title="RLAFFIN1:th.3">RLAFFIN1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">v</font> = <font color="Maroon" title="c14">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c6">cA</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_3_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a></span>;<br/><a NAME="E5:42_3_1_2"/><span class="lab"><font color="Green" title="E41">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></span>;<br/><a NAME="E6:42_3_1_2"/><span class="lab"><font color="Green" title="E42">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p3">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">fxAE</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c21">xESE</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_3_1_2"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/><a NAME="E8:42_3_1_2"/><span class="lab"><font color="Green" title="E43">A35</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">fxAE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_3_1_2"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:42_3_1_2"/><span class="lab"><font color="Green" title="E44">A36</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T65" title="RLAFFIN1:th.65">RLAFFIN1:65</a></span>;<br/><a NAME="E10:42_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A37</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E2:42_3_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a>, <a class="ref" href="rlaffin1.html#T77" title="RLAFFIN1:th.77">RLAFFIN1:77</a></span>;<br/><a NAME="E11:42_3_1_2"/><span class="lab"><font color="Green" title="E46">A38</font></span>: 
 the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a></span>;<br/><a NAME="E12:42_3_1_2"/><span class="lab"><font color="Green" title="E47">A39</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/><a NAME="E13:42_3_1_2"/>
 the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a>, <a class="ref" href="rlaffin1.html#T43" title="RLAFFIN1:th.43">RLAFFIN1:43</a></span>;<br/><a NAME="E14:42_3_1_2"/><span class="kw">then </span>
(  <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_3_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E9:42_3_1_2"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="txt" href="#E11:42_3_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E12:42_3_1_2"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="ref" href="rlaffin1.html#D7" title="RLAFFIN1:def.7">RLAFFIN1:def 7</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E15:42_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A40</font></span>: 
(  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p3">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="rlaffin1.html#T30" title="RLAFFIN1:th.30">RLAFFIN1:30</a>, <a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/><a NAME="E16:42_3_1_2"/><span class="lab"><font color="Green" title="E49">A41</font></span>: 
<font color="Maroon" title="c5">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#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E2:42_3_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E3:42_3_1_2"><span class="lab"><font color="Green" title="E40">A32</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="E17:42_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A42</font></span>: 
<font color="Maroon" title="c5">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#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a></span>;<br/><a NAME="E18:42_3_1_2"/><span class="kw">then </span>
 <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="rlaffin1.html#D7" title="RLAFFIN1:def.7">RLAFFIN1:def 7</a></span>;<br/><a NAME="E19:42_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A43</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c20">fxAE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c21">xESE</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E15:42_3_1_2"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="rlaffin1.html#T30" title="RLAFFIN1:th.30">RLAFFIN1:30</a></span>;<br/><a NAME="E20:42_3_1_2"/><span class="lab"><font color="Green" title="E52">A44</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">xESE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_3_1_2"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_2_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="E1:42_3_1_2_1"/>
(  ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon" title="c20">fxAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> ) or  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> holds <br/><font color="Maroon" title="c20">fxAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:42_3_1_2"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="rvsum_1.html#T83" title="RVSUM_1:th.83">RVSUM_1:83</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_3_1_2_1_1"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon" title="c20">fxAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:42_3_1_2_1_1"/><span class="lab"><font color="Green" title="E53">A45</font></span>: 
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:42_3_1_2_1_1"/><span class="lab"><font color="Green" title="E54">A46</font></span>: 
<font color="Maroon" title="c20">fxAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font>
 ;<br/><a NAME="E5:42_3_1_2_1_1"/><span class="lab"><font color="Green" title="E55">A47</font></span>: 
<font color="Maroon" title="c20">fxAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_3_1_2"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E6:42_3_1_2_1_1"/><span class="lab"><font color="Green" title="E56">A48</font></span>: 
<font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:42_3_1_2"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E7:42_3_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font></span>)</span>)</span>)</span> <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="#E2:42_3_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E10:42_3_1_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="ref" href="rlaffin3.html#D3" title="RLAFFIN3:def.3">RLAFFIN3:def 3</a></span>;<br/><a NAME="E8:42_3_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A49</font></span>: 
<span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:42_3_1_2"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E4:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E5:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="ref" href="rlaffin3.html#D3" title="RLAFFIN3:def.3">RLAFFIN3:def 3</a></span>;<br/><a NAME="E9:42_3_1_2_1_1"/><span class="lab"><font color="Green" title="E58">A50</font></span>: 
 the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_3_1_2"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><a NAME="E10:42_3_1_2_1_1"/><span class="kw">then </span>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E16:42_3_1_2"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E4:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E5:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E6:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="ref" href="rlaffin1.html#T71" title="RLAFFIN1:th.71">RLAFFIN1:71</a></span>;<br/><a NAME="E11:42_3_1_2_1_1"/><span class="kw">then </span>
 the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="ref" href="rlvect_2.html#T19" title="RLVECT_2:th.19">RLVECT_2:19</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:42_3_1_2_1_1"/><span class="lab"><font color="Green" title="E59">A51</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E14:42_3_1_2_1_1"/><span class="lab"><font color="Green" title="E60">A52</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">S</font>
 <span class="kw">and </span><br/><a NAME="E15:42_3_1_2_1_1"/><span class="lab"><font color="Green" title="E61">A53</font></span>: 
 the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_3_1_2"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><a NAME="E16:42_3_1_2_1_1"/>
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E13:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="txt" href="#E15:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E61">A53</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="E17:42_3_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A54</font></span>: 
<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E60">A52</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><a NAME="E18:42_3_1_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E53">A45</font></span></a></span>;<br/><a NAME="E19:42_3_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E20:42_3_1_2_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:42_3_1_2_1_1"><span class="lab"><font color="Green" title="E62">A54</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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="42_3_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_3_1_2_1_2"/><span class="lab"><font color="Green" title="E53">A55</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> holds <br/><font color="Maroon" title="c20">fxAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c22">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:42_3_1_2_1_2"/><span class="lab"><font color="Green" title="E54">A56</font></span>: 
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_1_2"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">j</font> = <font color="Maroon" title="c22">j</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a></span>;<br/><a NAME="E5:42_3_1_2_1_2"/><span class="lab"><font color="Green" title="E55">A57</font></span>: 
<font color="Maroon" title="c20">fxAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E1:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a></span>;<br/><a NAME="E6:42_3_1_2_1_2"/><span class="lab"><font color="Green" title="E56">A58</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c23">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a></span>;<br/><a NAME="E7:42_3_1_2_1_2"/>
<font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E20:42_3_1_2"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E8:42_3_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A59</font></span>: 
<font color="Maroon" title="c21">xESE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>)</span>)</span> <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="#E2:42_3_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E10:42_3_1_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="ref" href="rlaffin3.html#D3" title="RLAFFIN3:def.3">RLAFFIN3:def 3</a></span>;<br/><a NAME="E9:42_3_1_2_1_2"/>
<font color="Maroon" title="c20">fxAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E8:42_3_1_2"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E3:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E10:42_3_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:42_3_1_2"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E8:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E57">A59</font></span></a>, <a class="txt" href="#E5:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E55">A57</font></span></a>, <a class="ref" href="rlaffin3.html#D3" title="RLAFFIN3:def.3">RLAFFIN3:def 3</a></span>;<br/><a NAME="E11:42_3_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A60</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E56">A58</font></span></a></span>;<br/><a NAME="E12:42_3_1_2_1_2"/>
<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E13:42_3_1_2_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42_3_1_2_1_2"><span class="lab"><font color="Green" title="E58">A60</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E19:42"/><span class="kw">then </span>
 not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">F</font></span>)</span> 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="#E2:42"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="simplex2.html#T22" target="_self" title="SIMPLEX2:th.22">Th22</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E21:42"/><span class="lab"><font color="Green" title="E38">A61</font></span>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E22:42"/><span class="lab"><font color="Green" title="E39">A62</font></span>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">cA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E21:42"><span class="lab"><font color="Green" title="E38">A61</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">v</font> = <font color="Maroon" title="c12">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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/>
<span class="kw">set </span><font color="Maroon" title="c14">fv</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">vA</font> = <font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">fvA</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font>;<br/>
<a NAME="E24:42"/><span class="lab"><font color="Green" title="E40">A63</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></span>;<br/>
<a NAME="E25:42"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</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="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E21:42"><span class="lab"><font color="Green" title="E38">A61</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="E26:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A64</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">cA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a></span>;<br/>
<a NAME="E27:42"/><span class="kw">then </span>
(  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="rlaffin1.html#D7" title="RLAFFIN1:def.7">RLAFFIN1:def 7</a>, <a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<a NAME="E28:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A65</font></span>: 
(  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="rlaffin1.html#T30" title="RLAFFIN1:th.30">RLAFFIN1:30</a>, <a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<a NAME="E29:42"/><span class="lab"><font color="Green" title="E43">A66</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">fvAE</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c18">vAE</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/>
<a NAME="E31:42"/><span class="lab"><font color="Green" title="E44">A67</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">fvAE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E32:42"/><span class="lab"><font color="Green" title="E45">A68</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">vAE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:42"><span class="lab"><font color="Green" title="E43">A66</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E33:42"/><span class="lab"><font color="Green" title="E46">A69</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> holds <br/><font color="Maroon" title="c17">fvAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">vAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">j</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"> ( <font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font></span>)</span> implies <font color="Maroon" title="c17">fvAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">vAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_4"/><span class="lab"><font color="Green" title="E47">A70</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">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="c17">fvAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">vAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span><br/>

<a NAME="E2:42_4"/><span class="kw">then </span>
( <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">j</font>) &amp; <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</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="c11">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E37">A29</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="E3:42_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:42"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E4:42_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A71</font></span>: 
 ex <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c6">cA</font></span>)</span> st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> &amp; <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> )
 
;<br/>
<span class="lab"><font color="Green" title="E49">A72</font></span>: <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D3" title="RLAFFIN3:def.3">RLAFFIN3:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c18">vAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:42"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="txt" href="#E1:42_4"><span class="lab"><font color="Green" title="E47">A70</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:42"><span class="lab"><font color="Green" title="E41">A64</font></span></a>, <a class="ref" href="rlaffin3.html#D3" title="RLAFFIN3:def.3">RLAFFIN3:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c17">fvAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:42"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E1:42_4"><span class="lab"><font color="Green" title="E47">A70</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:42_4"/>
<font color="Maroon" title="c17">fvAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">vAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_4"><span class="lab"><font color="Green" title="E48">A71</font></span></a>, <a class="txt" href="#E5:42_4"><span class="lab"><font color="Green" title="E49">A72</font></span></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="E34:42"/><span class="lab"><font color="Green" title="E47">A73</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<a NAME="E35:42"/>
 <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E22:42"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="ref" href="rlaffin1.html#D7" title="RLAFFIN1:def.7">RLAFFIN1:def 7</a></span>;<br/>
<a NAME="E36:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A74</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c17">fvAE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c18">vAE</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E42">A65</font></span></a>, <a class="ref" href="rlaffin1.html#T30" title="RLAFFIN1:th.30">RLAFFIN1:30</a></span>;<br/>
<a NAME="E37:42"/><span class="lab"><font color="Green" title="E49">A75</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E50">A76</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_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="E1:42_5_1"/>
( <font color="Maroon" title="c19">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or  not <font color="Maroon" title="c19">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_1"/>
<font color="Maroon" title="c19">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:42_5_1_1"/><span class="lab"><font color="Green" title="E51">A77</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E4:42_5_1_1"/><span class="lab"><font color="Green" title="E52">A78</font></span>: 
 the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E36">A28</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="E5:42_5_1_1"/><span class="lab"><font color="Green" title="E53">A79</font></span>: 
( <font color="Maroon" title="c17">fvAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">w</font> &amp; <font color="Maroon" title="c18">vAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_5_1_1"><span class="lab"><font color="Green" title="E51">A77</font></span></a>, <a class="txt" href="#E4:42_5_1_1"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><a NAME="E6:42_5_1_1"/>
( <font color="Maroon" title="c17">fvAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">vAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c17">fvAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c18">vAE</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E36:42"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E33:42"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="txt" href="#E3:42_5_1_1"><span class="lab"><font color="Green" title="E51">A77</font></span></a>, <a class="ref" href="rvsum_1.html#T83" title="RVSUM_1:th.83">RVSUM_1:83</a></span>;<br/><span class="kw">hence </span><a NAME="E7:42_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_5_1_1"><span class="lab"><font color="Green" title="E53">A79</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_2"/><span class="lab"><font color="Green" title="E51">A80</font></span>: 
 not <font color="Maroon" title="c19">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:42_5_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c19">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:42"><span class="lab"><font color="Green" title="E47">A73</font></span></a></span>;<br/><a NAME="E3:42_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A81</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T19" title="RLVECT_2:th.19">RLVECT_2:19</a></span>;<br/><a NAME="E4:42_5_1_2"/>
 not <font color="Maroon" title="c19">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:42"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E1:42_5_1_2"><span class="lab"><font color="Green" title="E51">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_5_1_2"><span class="lab"><font color="Green" title="E52">A81</font></span></a>, <a class="ref" href="rlvect_2.html#T19" title="RLVECT_2:th.19">RLVECT_2:19</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><span class="kw">end;</span></div>
<a NAME="E39:42"/><span class="lab"><font color="Green" title="E51">A82</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E22:42"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="ref" href="rlaffin1.html#D7" title="RLAFFIN1:def.7">RLAFFIN1:def 7</a></span>;<br/>
<a NAME="E40:42"/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E26:42"><span class="lab"><font color="Green" title="E41">A64</font></span></a>, <a class="ref" href="rlaffin1.html#D7" title="RLAFFIN1:def.7">RLAFFIN1:def 7</a></span>;<br/>
<a NAME="E41:42"/><span class="kw">then </span>
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:42"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="txt" href="#E39:42"><span class="lab"><font color="Green" title="E51">A82</font></span></a>, <a class="ref" href="rlvect_2.html#D9" title="RLVECT_2:def.9">RLVECT_2:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E42:42"/>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E21:42"><span class="lab"><font color="Green" title="E38">A61</font></span></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>
