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

<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="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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a>  implies <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c3">A</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<a NAME="E2:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <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="rlaffin1.html#T50" onmouseover="rs('rlaffin1/T50')" onmouseout="rh()">RLAFFIN1:50</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1_1"/>
( <font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c3">A</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_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_1"/>
<font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_1_1_1_1"/>
<font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2"/><span class="lab"><font color="Green" title="E34">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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><div class="add"><a NAME="E2:42_1_1_1_2"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">Ia</font> being   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:42_1_1_1_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Ia</font>
 <span class="kw">and </span><br/><a NAME="E5:42_1_1_1_2"/>
<font color="Maroon" title="c4">Ia</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E6:42_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">Ia</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="txt" href="#E2:42_1_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="rlaffin1.html#T60" onmouseover="rs('rlaffin1/T60')" onmouseout="rh()">RLAFFIN1:60</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">IA</font> being   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:42_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c4">Ia</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">IA</font>
 <span class="kw">and </span><br/><a NAME="E9:42_1_1_1_2"/>
<font color="Maroon" title="c5">IA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:42_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c5">IA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T60" onmouseover="rs('rlaffin1/T60')" onmouseout="rh()">RLAFFIN1:60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">IB</font> = <font color="Maroon" title="c5">IA</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">Ia</font> as   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T43" onmouseover="rs('rlaffin1/T43')" onmouseout="rh()">RLAFFIN1:43</a>, <a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">cIB</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font>;<br/><a NAME="E12:42_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T4" target="_self" onmouseover="rs('rlaffin3/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E13:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T5" target="_self" onmouseover="rs('rlaffin3/T5')" onmouseout="rh()">Th5</a></span>;<br/><a NAME="E14:42_1_1_1_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_4.html#T22" onmouseover="rs('rusub_4/T22')" onmouseout="rh()">RUSUB_4:22</a></span>;<br/><a NAME="E15:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A8</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T50" onmouseover="rs('rlaffin1/T50')" onmouseout="rh()">RLAFFIN1:50</a></span>;<br/><a NAME="E16:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A9</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">IA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1_1_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E12:42_1_1_1_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="rlaffin3.html#T6" target="_self" onmouseover="rs('rlaffin3/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E17:42_1_1_1_2"/>
 not <font color="Maroon" title="c4">Ia</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_1_1_1_2"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E35">A3</font></span></a></span>;<br/><a NAME="E18:42_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">IA</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">Ia</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_1_1_1_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="xboole_1.html#T67" onmouseover="rs('xboole_1/T67')" onmouseout="rh()">XBOOLE_1:67</a></span>;<br/><a NAME="E19:42_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">IA</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">IB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T83" onmouseover="rs('xboole_1/T83')" onmouseout="rh()">XBOOLE_1:83</a></span>;<br/><a NAME="E20:42_1_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="card_2.html#T102" onmouseover="rs('card_2/T102')" onmouseout="rh()">CARD_2:102</a>, <a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/><a NAME="E21:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A10</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42_1_1_1_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E22:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A11</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">IA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1_1_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E15:42_1_1_1_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E12:42_1_1_1_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="rlaffin3.html#T6" target="_self" onmouseover="rs('rlaffin3/T6')" onmouseout="rh()">Th6</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">TRc</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span>;<br/><a NAME="E23:42_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T66" onmouseover="rs('euclid/T66')" onmouseout="rh()">EUCLID:66</a></span>;<br/><a NAME="E24:42_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> is   <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> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D4" onmouseover="rs('euclid/D4')" onmouseout="rh()">EUCLID:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">P</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E26:42_1_1_1_2"/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D4" onmouseover="rs('euclid/D4')" onmouseout="rh()">EUCLID:def 4</a></span>;<br/><a NAME="E27:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<font color="Maroon" title="c9">P</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="#E23:42_1_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="pcomps_1.html#T7" onmouseover="rs('pcomps_1/T7')" onmouseout="rh()">PCOMPS_1:7</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">P1</font> =  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c4">Ia</font>;<br/><a NAME="E28:42_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A14</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="c4">Ia</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Ia</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" target="_self" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">Def1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">P2</font> =  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c6">IB</font>;<br/><span class="kw">set </span><font color="Maroon" title="c12">P12</font> =  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c6">IB</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c4">Ia</font>;<br/><a NAME="E29:42_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A15</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="c6">IB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" target="_self" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E30:42_1_1_1_2"/>
<font color="Maroon" title="c4">Ia</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">IB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>;<br/><a NAME="E31:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</font></span>: 
 the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c6">IB</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c4">Ia</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42_1_1_1_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="finseq_3.html#T91" onmouseover="rs('finseq_3/T91')" onmouseout="rh()">FINSEQ_3:91</a></span>;<br/><font color="Maroon" title="c4">Ia</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">IB</font> = 
<font color="Maroon" title="c4">Ia</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">IA</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" onmouseover="rs('xboole_1/T39')" onmouseout="rh()">XBOOLE_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c5">IA</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_1_1_1_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>
;<br/><a NAME="E33:42_1_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c6">IB</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c4">Ia</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42_1_1_1_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="finseq_1.html#T31" onmouseover="rs('finseq_1/T31')" onmouseout="rh()">FINSEQ_1:31</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">P12</font> =  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c6">IB</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a>  the    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c4">Ia</font> as    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c5">IA</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:42_1_1_1_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="rlaffin3.html#D1" target="_self" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E35:42_1_1_1_2"/>
 <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="c6">IB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="finseq_4.html#T62" onmouseover="rs('finseq_4/T62')" onmouseout="rh()">FINSEQ_4:62</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A17</font></span>: <font color="Maroon" title="c13">P12</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c13">P12</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><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="c6">IB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P12</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><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="c6">IB</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T115" onmouseover="rs('relat_1/T115')" onmouseout="rh()">RELAT_1:115</a></span>
<br/>.= 
 <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="c6">IB</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T21" onmouseover="rs('finseq_1/T21')" onmouseout="rh()">FINSEQ_1:21</a></span>
<br/>.= 
<font color="Maroon" title="c6">IB</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" target="_self" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">Def1</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c14">B</font> = <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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span> ;<br/><span class="lab"><font color="Green" title="E50">A18</font></span>: <font color="Maroon" title="c5">IA</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">IB</font> = 
<font color="Maroon" title="c5">IA</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Ia</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T48" onmouseover="rs('xboole_1/T48')" onmouseout="rh()">XBOOLE_1:48</a></span>
<br/>.= 
<font color="Maroon" title="c4">Ia</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_1_1_1_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/><a NAME="E38:42_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A19</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_4"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c15">x</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="c17">vP</font> = <font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font>;<br/>
<a NAME="E3:42_1_1_1_2_4"/><span class="lab"><font color="Green" title="E53">A21</font></span>: 
<font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</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="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T8" onmouseover="rs('rfinseq/T8')" onmouseout="rh()">RFINSEQ:8</a></span>;<br/>
<a NAME="E4:42_1_1_1_2_4"/>
<font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E10:42_1_1_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E15:42_1_1_1_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E37:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="txt" href="#E36:42_1_1_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="txt" href="#E22:42_1_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E1:42_1_1_1_2_4"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="rlaffin3.html#T22" target="_self" onmouseover="rs('rlaffin3/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E5:42_1_1_1_2_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_4"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>;<br/>
<a NAME="E6:42_1_1_1_2_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:42_1_1_1_2_4"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E39:42_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A22</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:42_1_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E40:42_1_1_1_2"/>
<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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</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> <font color="Maroon" title="c3">A</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_5"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">v</font> being   <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><span class="kw"> such that </span><br/><a NAME="E3:42_1_1_1_2_5"/><span class="lab"><font color="Green" title="E53">A23</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v</font>
 <span class="kw">and </span><br/><a NAME="E4:42_1_1_1_2_5"/><span class="lab"><font color="Green" title="E54">A24</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">vP</font> = <font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font>;<br/>
<a NAME="E5:42_1_1_1_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1_1_2_5"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:42_1_1_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T8" onmouseover="rs('rfinseq/T8')" onmouseout="rh()">RFINSEQ:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:42_1_1_1_2_5"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="#E2:42_1_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E10:42_1_1_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E15:42_1_1_1_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E37:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="txt" href="#E36:42_1_1_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="txt" href="#E22:42_1_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E3:42_1_1_1_2_5"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="ref" href="rlaffin3.html#T22" target="_self" onmouseover="rs('rlaffin3/T22')" onmouseout="rh()">Th22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E41:42_1_1_1_2"/><span class="kw">then </span>
<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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c13">P12</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">IB</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="#E38:42_1_1_1_2"><span class="lab"><font color="Green" title="E51">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E42:42_1_1_1_2"/>
<font color="Maroon" title="c3">A</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="#E2:42_1_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E39:42_1_1_1_2"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E27:42_1_1_1_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="rlaffin3.html#T26" target="_self" onmouseover="rs('rlaffin3/T26')" onmouseout="rh()">Th26</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
