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

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">X</font> holds <br/> <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">X</font> is  <a href="incproj.html#V9" title="INCPROJ:attr.9">Desarguesian</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> implies  <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font> is  <a href="incproj.html#V9" title="INCPROJ:attr.9">Desarguesian</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font> is  <a href="incproj.html#V9" title="INCPROJ:attr.9">Desarguesian</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">b2</font>, <font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c9">a3</font>, <font color="Maroon" title="c10">r</font>, <font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c12">t</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="incproj.html#D19" title="INCPROJ:def.19">INCPROJ:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>  holds <br/>(  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>1</sub></font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>2</sub></font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>3</sub></font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>4</sub></font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>5</sub></font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>6</sub></font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c6">b2</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>7</sub></font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>8</sub></font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>9</sub></font> or  not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a1</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a3</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font> or <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c7">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> or <font color="Maroon" title="c9">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font> or  ex <font color="Olive">b<sub>10</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>10</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">C1</font>, <font color="Maroon" title="c14">C2</font>, <font color="Maroon" title="c15">C3</font>, <font color="Maroon" title="c16">A1</font>, <font color="Maroon" title="c17">A2</font>, <font color="Maroon" title="c18">A3</font>, <font color="Maroon" title="c19">B1</font>, <font color="Maroon" title="c20">B2</font>, <font color="Maroon" title="c21">B3</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">C1</font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">C2</font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">C3</font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">A1</font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">A2</font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">A3</font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c6">b2</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c19">B1</font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c20">B2</font> or  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">B3</font> or  not <font color="Maroon" title="c13">C1</font>,<font color="Maroon" title="c14">C2</font>,<font color="Maroon" title="c15">C3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a1</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a3</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font> or <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c7">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> or <font color="Maroon" title="c9">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">C1</font>
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">C2</font>
 <span class="kw">and </span><br/><a NAME="E5:16"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">C3</font>
 <span class="kw">and </span><br/><a NAME="E6:16"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">A1</font>
 <span class="kw">and </span><br/><a NAME="E7:16"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">A2</font>
 <span class="kw">and </span><br/><a NAME="E8:16"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">A3</font>
 <span class="kw">and </span><br/><a NAME="E9:16"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c6">b2</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c19">B1</font>
 <span class="kw">and </span><br/><a NAME="E10:16"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c20">B2</font>
 <span class="kw">and </span><br/><a NAME="E11:16"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">B3</font>
 <span class="kw">and </span><br/><a NAME="E12:16"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c13">C1</font>,<font color="Maroon" title="c14">C2</font>,<font color="Maroon" title="c15">C3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <span class="kw">and </span><br/><a NAME="E13:16"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a1</font>
 <span class="kw">and </span><br/><a NAME="E14:16"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a3</font> )
 <span class="kw">and </span><br/><a NAME="E15:16"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b1</font>
 <span class="kw">and </span><br/><a NAME="E16:16"/><span class="lab"><font color="Green" title="E29">A16</font></span>: 
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b3</font> )
 <span class="kw">and </span><br/><a NAME="E17:16"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c5">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b1</font>
 <span class="kw">and </span><br/><a NAME="E18:16"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c7">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b2</font>
 <span class="kw">and </span><br/><a NAME="E19:16"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
<font color="Maroon" title="c9">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<a NAME="E20:16"/><span class="lab"><font color="Green" title="E33">A20</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E21:16"/><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c6">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E22:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A22</font></span>: 
<font color="Maroon" title="c6">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E23:16"/><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c7">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E24:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A24</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E25:16"/><span class="lab"><font color="Green" title="E38">A25</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E26:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A26</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E27:16"/><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E28:16"/><span class="lab"><font color="Green" title="E41">A28</font></span>: 
( <font color="Maroon" title="c9">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A1</font> &amp; <font color="Maroon" title="c7">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E29:16"/><span class="lab"><font color="Green" title="E42">A29</font></span>: 
<font color="Maroon" title="c8">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E30:16"/><span class="lab"><font color="Green" title="E43">A30</font></span>: 
<font color="Maroon" title="c8">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E31:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A31</font></span>: 
<font color="Maroon" title="c8">b3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E32:16"/><span class="lab"><font color="Green" title="E45">A32</font></span>: 
<font color="Maroon" title="c9">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E33:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E34:16"/><span class="lab"><font color="Green" title="E47">A34</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E35:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A35</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E36:16"/><span class="lab"><font color="Green" title="E49">A36</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">a3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:16"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E37:16"/><span class="lab"><font color="Green" title="E50">A37</font></span>: 
( <font color="Maroon" title="c9">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A2</font> &amp; <font color="Maroon" title="c5">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E38:16"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
<font color="Maroon" title="c4">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E39:16"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c13">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<a NAME="E40:16"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c4">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E41:16"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
<font color="Maroon" title="c14">C2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<a NAME="E42:16"/><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c8">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E43:16"/><span class="lab"><font color="Green" title="E56">A43</font></span>: 
<font color="Maroon" title="c13">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<a NAME="E44:16"/><span class="lab"><font color="Green" title="E57">A44</font></span>: 
<font color="Maroon" title="c6">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E45:16"/><span class="lab"><font color="Green" title="E58">A45</font></span>: 
<font color="Maroon" title="c5">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E46:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A46</font></span>: 
<font color="Maroon" title="c5">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E47:16"/><span class="lab"><font color="Green" title="E60">A47</font></span>: 
<font color="Maroon" title="c4">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E48:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A48</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">b1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E49:16"/><span class="lab"><font color="Green" title="E62">A49</font></span>: 
<font color="Maroon" title="c6">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E50:16"/><span class="lab"><font color="Green" title="E63">A50</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:16"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="txt" href="#E45:16"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E51:16"/><span class="lab"><font color="Green" title="E64">A51</font></span>: 
( not <font color="Maroon" title="c5">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B2</font> &amp; not <font color="Maroon" title="c7">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> &amp; not <font color="Maroon" title="c9">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1"/>
( <font color="Maroon" title="c5">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B2</font> or <font color="Maroon" title="c7">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> or <font color="Maroon" title="c9">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1"/><span class="kw">then </span>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c20">B2</font> or <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">B3</font> or <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c19">B1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E40:16"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E49:16"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E3:16_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C1</font> or <font color="Maroon" title="c4">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C2</font> or <font color="Maroon" title="c6">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E44:16"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E38:16"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E50:16"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E24:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E33:16"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="incsp_1.html#D10" title="INCSP_1:def.10">INCSP_1:def 10</a></span>;<br/>
<a NAME="E4:16_1"/><span class="kw">then </span>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">C1</font> or <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">b1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">C2</font> or <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">C3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E34:16"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E48:16"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E35:16"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E43:16"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E41:16"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E39:16"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="incsp_1.html#D10" title="INCSP_1:def.10">INCSP_1:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E52:16"/><span class="lab"><font color="Green" title="E65">A52</font></span>: 
( <font color="Maroon" title="c11">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A3</font> &amp; <font color="Maroon" title="c11">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E53:16"/><span class="lab"><font color="Green" title="E66">A53</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E54:16"/><span class="lab"><font color="Green" title="E67">A54</font></span>: 
( <font color="Maroon" title="c10">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A2</font> &amp; <font color="Maroon" title="c10">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E55:16"/><span class="lab"><font color="Green" title="E68">A55</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E56:16"/><span class="lab"><font color="Green" title="E69">A56</font></span>: 
<font color="Maroon" title="c5">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E57:16"/><span class="lab"><font color="Green" title="E70">A57</font></span>: 
<font color="Maroon" title="c7">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E58:16"/><span class="lab"><font color="Green" title="E71">A58</font></span>: 
the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E59:16"/><span class="lab"><font color="Green" title="E72">A59</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E45:16"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E60:16"/><span class="lab"><font color="Green" title="E73">A60</font></span>: 
( not <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A1</font> &amp; not <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B1</font> &amp; not <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A2</font> &amp; not <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B2</font> &amp; not <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A3</font> &amp; not <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2"/>
( <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A1</font> or <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B1</font> or <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A2</font> or <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B2</font> or <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A3</font> or <font color="Maroon" title="c3">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2"/><span class="kw">then </span>
( ( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">A1</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">a3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">A1</font> ) or ( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c19">B1</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c19">B1</font> ) or ( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">A2</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">a3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">A2</font> ) or ( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">b1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c20">B2</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c20">B2</font> ) or ( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">A3</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">a1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">A3</font> ) or ( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">B3</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">b1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">B3</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="txt" href="#E37:16"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E57:16"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="txt" href="#E56:16"><span class="lab"><font color="Green" title="E69">A56</font></span></a>, <a class="txt" href="#E44:16"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E40:16"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E38:16"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E49:16"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E3:16_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c16">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C2</font> &amp; <font color="Maroon" title="c16">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">C3</font> ) or ( <font color="Maroon" title="c19">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C2</font> &amp; <font color="Maroon" title="c19">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">C3</font> ) or ( <font color="Maroon" title="c17">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C1</font> &amp; <font color="Maroon" title="c17">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">C3</font> ) or ( <font color="Maroon" title="c20">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C1</font> &amp; <font color="Maroon" title="c20">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">C3</font> ) or ( <font color="Maroon" title="c18">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C2</font> &amp; <font color="Maroon" title="c18">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C1</font> ) or ( <font color="Maroon" title="c21">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C2</font> &amp; <font color="Maroon" title="c21">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E59:16"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E36:16"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E48:16"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E35:16"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="ref" href="incsp_1.html#D10" title="INCSP_1:def.10">INCSP_1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:16_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">consider </span><font color="Maroon" title="c22">salpha</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E62:16"/><span class="lab"><font color="Green" title="E74">A61</font></span>: 
( <font color="Maroon" title="c22">salpha</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A3</font> &amp; <font color="Maroon" title="c22">salpha</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> )
 <span class="kw">and </span><br/><a NAME="E63:16"/><span class="lab"><font color="Green" title="E75">A62</font></span>: 
<font color="Maroon" title="c22">salpha</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E47:16"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="txt" href="#E45:16"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E21:16"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E57:16"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="txt" href="#E56:16"><span class="lab"><font color="Green" title="E69">A56</font></span></a>, <a class="txt" href="#E38:16"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E49:16"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E43:16"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E51:16"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="ref" href="combgras.html#T12" target="_self" title="COMBGRAS:th.12">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">ralpha</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E65:16"/><span class="lab"><font color="Green" title="E76">A63</font></span>: 
( <font color="Maroon" title="c23">ralpha</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B2</font> &amp; <font color="Maroon" title="c23">ralpha</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A2</font> )
 <span class="kw">and </span><br/><a NAME="E66:16"/><span class="lab"><font color="Green" title="E77">A64</font></span>: 
<font color="Maroon" title="c23">ralpha</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E47:16"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="txt" href="#E45:16"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E34:16"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E30:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E37:16"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E40:16"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E39:16"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E51:16"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E60:16"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="ref" href="combgras.html#T12" target="_self" title="COMBGRAS:th.12">Th12</a></span>;<br/>
<a NAME="E67:16"/><span class="lab"><font color="Green" title="E78">A65</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">talpha</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E69:16"/><span class="lab"><font color="Green" title="E79">A66</font></span>: 
( <font color="Maroon" title="c24">talpha</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A1</font> &amp; <font color="Maroon" title="c24">talpha</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B1</font> )
 <span class="kw">and </span><br/><a NAME="E70:16"/><span class="lab"><font color="Green" title="E80">A67</font></span>: 
<font color="Maroon" title="c24">talpha</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E21:16"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E34:16"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E30:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="txt" href="#E44:16"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E41:16"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E51:16"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E60:16"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="ref" href="combgras.html#T12" target="_self" title="COMBGRAS:th.12">Th12</a></span>;<br/>
<a NAME="E71:16"/><span class="lab"><font color="Green" title="E81">A68</font></span>: 
( <font color="Maroon" title="c16">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">B1</font> &amp; <font color="Maroon" title="c17">A2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">B2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E51:16"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E72:16"/><span class="lab"><font color="Green" title="E82">A69</font></span>: 
( <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">ralpha</font> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">salpha</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">talpha</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_3"/><span class="lab"><font color="Green" title="E83">A70</font></span>: 
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c22">salpha</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">A3</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c22">salpha</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">B3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:16"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E62:16"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E2:16_3"/><span class="lab"><font color="Green" title="E84">A71</font></span>: 
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c23">ralpha</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">A2</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c23">ralpha</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c20">B2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:16"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="txt" href="#E65:16"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:16_3"/><span class="lab"><font color="Green" title="E85">A72</font></span>: 
( <font color="Maroon" title="c10">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">ralpha</font> or <font color="Maroon" title="c11">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">salpha</font> or <font color="Maroon" title="c12">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">talpha</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:16_3"/>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c24">talpha</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">A1</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c24">talpha</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c19">B1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:16"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E53:16"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="txt" href="#E69:16"><span class="lab"><font color="Green" title="E79">A66</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:16"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="txt" href="#E51:16"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E71:16"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="txt" href="#E3:16_3"><span class="lab"><font color="Green" title="E85">A72</font></span></a>, <a class="txt" href="#E2:16_3"><span class="lab"><font color="Green" title="E84">A71</font></span></a>, <a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E83">A70</font></span></a>, <a class="ref" href="incsp_1.html#D10" title="INCSP_1:def.10">INCSP_1:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E73:16"/><span class="kw">then </span>
<font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</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="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:16"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E66:16"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<a NAME="E74:16"/><span class="kw">then </span>
<font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</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="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<a NAME="E75:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A73</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</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">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:16"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E72:16"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="#E67:16"><span class="lab"><font color="Green" title="E78">A65</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E76:16"/>
<font color="Maroon" title="c7">a2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E77:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A74</font></span>: 
<font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E78:16"/>
<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E79:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A75</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<a NAME="E80:16"/>
<font color="Maroon" title="c13">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E81:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A76</font></span>: 
 ex <font color="Olive" title="b1">C11</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">C11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:16"><span class="lab"><font color="Green" title="E71">A58</font></span></a></span>;<br/>
<a NAME="E82:16"/><span class="lab"><font color="Green" title="E87">A77</font></span>: 
<font color="Maroon" title="c4">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E47:16"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E83:16"/><span class="kw">then </span>
<font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E84:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A78</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:16"><span class="lab"><font color="Green" title="E86">A76</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<a NAME="E85:16"/><span class="lab"><font color="Green" title="E89">A79</font></span>: 
the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E86:16"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E87:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A80</font></span>: 
 ex <font color="Olive" title="b1">o1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E88:16"/>
<font color="Maroon" title="c4">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E89:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A81</font></span>: 
 ex <font color="Olive" title="b1">b11</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">b11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">b11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E90:16"/>
<font color="Maroon" title="c9">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E91:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A82</font></span>: 
 ex <font color="Olive" title="b1">a13</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">a13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a3</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">a13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E92:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A83</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
;<br/>
<a NAME="E93:16"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E94:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A84</font></span>: 
 ex <font color="Olive" title="b1">t1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E95:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A85</font></span>: 
<font color="Maroon" title="c12">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E96:16"/>
<font color="Maroon" title="c7">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E97:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A86</font></span>: 
 ex <font color="Olive" title="b1">a12</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">a12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">a12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E98:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A87</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
;<br/>
<a NAME="E99:16"/>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E100:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A88</font></span>: 
 ex <font color="Olive" title="b1">s1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E101:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A89</font></span>: 
<font color="Maroon" title="c11">s</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E102:16"/>
<font color="Maroon" title="c5">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E103:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A90</font></span>: 
 ex <font color="Olive" title="b1">a11</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">a11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">a11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E104:16"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:16"><span class="lab"><font color="Green" title="E91">A81</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E105:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A91</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:16"><span class="lab"><font color="Green" title="E88">A78</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E106:16"/><span class="lab"><font color="Green" title="E102">A92</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/>
<a NAME="E107:16"/>
<font color="Maroon" title="c14">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E108:16"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C12</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">C12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:16"><span class="lab"><font color="Green" title="E71">A58</font></span></a></span>;<br/>
<a NAME="E109:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A93</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:16"><span class="lab"><font color="Green" title="E84">A74</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<a NAME="E110:16"/>
<font color="Maroon" title="c6">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E111:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A94</font></span>: 
 ex <font color="Olive" title="b1">b12</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">b12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">b12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E112:16"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E97:16"><span class="lab"><font color="Green" title="E96">A86</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E113:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A95</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E103">A93</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E114:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A96</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="txt" href="#E111:16"><span class="lab"><font color="Green" title="E104">A94</font></span></a>, <a class="txt" href="#E98:16"><span class="lab"><font color="Green" title="E97">A87</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E115:16"/><span class="lab"><font color="Green" title="E107">A97</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="txt" href="#E111:16"><span class="lab"><font color="Green" title="E104">A94</font></span></a>, <a class="txt" href="#E113:16"><span class="lab"><font color="Green" title="E105">A95</font></span></a>, <a class="txt" href="#E98:16"><span class="lab"><font color="Green" title="E97">A87</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E116:16"/><span class="lab"><font color="Green" title="E108">A98</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:16"><span class="lab"><font color="Green" title="E100">A90</font></span></a></span>;<br/>
<a NAME="E117:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A99</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="txt" href="#E89:16"><span class="lab"><font color="Green" title="E91">A81</font></span></a>, <a class="txt" href="#E105:16"><span class="lab"><font color="Green" title="E101">A91</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E118:16"/>
<font color="Maroon" title="c9">a3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E119:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A100</font></span>: 
<font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">b3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:16"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E120:16"/>
<font color="Maroon" title="c11">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E121:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A101</font></span>: 
<font color="Maroon" title="c11">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<a NAME="E122:16"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E123:16"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E124:16"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/>
<a NAME="E125:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A102</font></span>: 
( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E126:16"/><span class="lab"><font color="Green" title="E113">A103</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E127:16"/><span class="lab"><font color="Green" title="E114">A104</font></span>: 
not <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_4"/><span class="lab"><font color="Green" title="E115">A105</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_4"/><span class="lab"><font color="Green" title="E116">A106</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c25">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:16"><span class="lab"><font color="Green" title="E90">A80</font></span></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">z3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:16_4"/><span class="lab"><font color="Green" title="E117">A107</font></span>: 
<font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c26">z3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:16"><span class="lab"><font color="Green" title="E91">A81</font></span></a>, <a class="txt" href="#E1:16_4"><span class="lab"><font color="Green" title="E115">A105</font></span></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:16_4"/><span class="lab"><font color="Green" title="E118">A108</font></span>: 
<font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c27">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:16"><span class="lab"><font color="Green" title="E100">A90</font></span></a>, <a class="txt" href="#E1:16_4"><span class="lab"><font color="Green" title="E115">A105</font></span></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></span>;<br/>
<a NAME="E8:16_4"/>
<font color="Maroon" title="c3">o</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E126:16"><span class="lab"><font color="Green" title="E113">A103</font></span></a>, <a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E9:16_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">a1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:16"><span class="lab"><font color="Green" title="E87">A77</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E10:16_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c25">z1</font>,<font color="Maroon" title="c27">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_4"><span class="lab"><font color="Green" title="E116">A106</font></span></a>, <a class="txt" href="#E7:16_4"><span class="lab"><font color="Green" title="E118">A108</font></span></a>, <a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>;<br/>
<a NAME="E11:16_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A109</font></span>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c25">z1</font>,<font color="Maroon" title="c27">z2</font>,<font color="Maroon" title="c26">z3</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_4"><span class="lab"><font color="Green" title="E117">A107</font></span></a>, <a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></span>;<br/>
<a NAME="E12:16_4"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c25">z1</font>,<font color="Maroon" title="c27">z2</font>,<font color="Maroon" title="c26">z3</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E3:16_4"><span class="lab"><font color="Green" title="E116">A106</font></span></a>, <a class="txt" href="#E7:16_4"><span class="lab"><font color="Green" title="E118">A108</font></span></a>, <a class="txt" href="#E5:16_4"><span class="lab"><font color="Green" title="E117">A107</font></span></a>, <a class="ref" href="card_2.html#T77" title="CARD_2:th.77">CARD_2:77</a></span>;<br/>
<a NAME="E13:16_4"/><span class="kw">then </span>
3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c13">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_4"><span class="lab"><font color="Green" title="E119">A109</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:16_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:16"><span class="lab"><font color="Green" title="E86">A76</font></span></a>, <a class="txt" href="#E1:16_4"><span class="lab"><font color="Green" title="E115">A105</font></span></a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E128:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A110</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E125:16"><span class="lab"><font color="Green" title="E112">A102</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<a NAME="E129:16"/>
<font color="Maroon" title="c15">C3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E130:16"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C13</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">C13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">C3</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:16"><span class="lab"><font color="Green" title="E71">A58</font></span></a></span>;<br/>
<a NAME="E131:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A111</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">b3</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E119:16"><span class="lab"><font color="Green" title="E110">A100</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<a NAME="E132:16"/>
<font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E133:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A112</font></span>: 
 ex <font color="Olive" title="b1">b13</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">b13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">b13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E134:16"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">b3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E91:16"><span class="lab"><font color="Green" title="E92">A82</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E135:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A113</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E131:16"><span class="lab"><font color="Green" title="E116">A111</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E136:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A114</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="txt" href="#E133:16"><span class="lab"><font color="Green" title="E117">A112</font></span></a>, <a class="txt" href="#E92:16"><span class="lab"><font color="Green" title="E93">A83</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E137:16"/>
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E138:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A115</font></span>: 
 ex <font color="Olive" title="b1">r1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:16"><span class="lab"><font color="Green" title="E89">A79</font></span></a></span>;<br/>
<a NAME="E139:16"/><span class="kw">then </span>
<font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E100:16"><span class="lab"><font color="Green" title="E98">A88</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E140:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A116</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:16"><span class="lab"><font color="Green" title="E94">A84</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E141:16"/><span class="lab"><font color="Green" title="E122">A117</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="txt" href="#E133:16"><span class="lab"><font color="Green" title="E117">A112</font></span></a>, <a class="txt" href="#E135:16"><span class="lab"><font color="Green" title="E118">A113</font></span></a>, <a class="txt" href="#E92:16"><span class="lab"><font color="Green" title="E93">A83</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E142:16"/><span class="lab"><font color="Green" title="E123">A118</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="txt" href="#E89:16"><span class="lab"><font color="Green" title="E91">A81</font></span></a>, <a class="txt" href="#E105:16"><span class="lab"><font color="Green" title="E101">A91</font></span></a>, <a class="txt" href="#E116:16"><span class="lab"><font color="Green" title="E108">A98</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E143:16"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:16"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E72:16"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="#E100:16"><span class="lab"><font color="Green" title="E98">A88</font></span></a></span>;<br/>
<a NAME="E144:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A119</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E128:16"><span class="lab"><font color="Green" title="E115">A110</font></span></a>, <a class="txt" href="#E142:16"><span class="lab"><font color="Green" title="E123">A118</font></span></a>, <a class="txt" href="#E115:16"><span class="lab"><font color="Green" title="E107">A97</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E145:16"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:16"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E72:16"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="#E94:16"><span class="lab"><font color="Green" title="E94">A84</font></span></a></span>;<br/>
<a NAME="E146:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A120</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E128:16"><span class="lab"><font color="Green" title="E115">A110</font></span></a>, <a class="txt" href="#E115:16"><span class="lab"><font color="Green" title="E107">A97</font></span></a>, <a class="txt" href="#E141:16"><span class="lab"><font color="Green" title="E122">A117</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E147:16"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:16"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="txt" href="#E72:16"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="#E138:16"><span class="lab"><font color="Green" title="E120">A115</font></span></a></span>;<br/>
<a NAME="E148:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A121</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">b3</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E128:16"><span class="lab"><font color="Green" title="E115">A110</font></span></a>, <a class="txt" href="#E142:16"><span class="lab"><font color="Green" title="E123">A118</font></span></a>, <a class="txt" href="#E141:16"><span class="lab"><font color="Green" title="E122">A117</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E149:16"/><span class="lab"><font color="Green" title="E127">A122</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E150:16"/><span class="lab"><font color="Green" title="E128">A123</font></span>: 
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies  ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_5"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font></span><br/>

<a NAME="E2:16_5"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E117:16"><span class="lab"><font color="Green" title="E109">A99</font></span></a>, <a class="txt" href="#E114:16"><span class="lab"><font color="Green" title="E106">A96</font></span></a>, <a class="txt" href="#E136:16"><span class="lab"><font color="Green" title="E119">A114</font></span></a>, <a class="txt" href="#E75:16"><span class="lab"><font color="Green" title="E83">A73</font></span></a>, <a class="txt" href="#E148:16"><span class="lab"><font color="Green" title="E126">A121</font></span></a>, <a class="txt" href="#E146:16"><span class="lab"><font color="Green" title="E125">A120</font></span></a>, <a class="txt" href="#E144:16"><span class="lab"><font color="Green" title="E124">A119</font></span></a>, <a class="ref" href="combgras.html#T7" target="_self" title="COMBGRAS:th.7">Th7</a></span>;<br/>
<a NAME="E3:16_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:16"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E140:16"><span class="lab"><font color="Green" title="E121">A116</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">O</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:16_5"/><span class="lab"><font color="Green" title="E129">A124</font></span>: 
<font color="Maroon" title="c25">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font>
 ;<br/>
<a NAME="E6:16_5"/><span class="lab"><font color="Green" title="E130">A125</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E149:16"><span class="lab"><font color="Green" title="E127">A122</font></span></a>, <a class="txt" href="#E5:16_5"><span class="lab"><font color="Green" title="E129">A124</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E7:16_5"/>
( <font color="Maroon" title="c10">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> &amp; <font color="Maroon" title="c11">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E79:16"><span class="lab"><font color="Green" title="E85">A75</font></span></a>, <a class="txt" href="#E121:16"><span class="lab"><font color="Green" title="E111">A101</font></span></a>, <a class="txt" href="#E5:16_5"><span class="lab"><font color="Green" title="E129">A124</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E8:16_5"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c25">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_5"><span class="lab"><font color="Green" title="E130">A125</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:16_5"/>
 ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E151:16"/><span class="lab"><font color="Green" title="E129">A126</font></span>: 
<font color="Maroon" title="c10">r</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E138:16"><span class="lab"><font color="Green" title="E120">A115</font></span></a></span>;<br/>
<a NAME="E152:16"/><span class="lab"><font color="Green" title="E130">A127</font></span>: 
( 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> implies  ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_6"/><span class="lab"><font color="Green" title="E131">A128</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_6_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font></span><br/>

<a NAME="E2:16_6_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:16"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E140:16"><span class="lab"><font color="Green" title="E121">A116</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">O</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:16_6_1"/><span class="lab"><font color="Green" title="E132">A129</font></span>: 
<font color="Maroon" title="c25">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font>
 ;<br/>
<a NAME="E5:16_6_1"/><span class="lab"><font color="Green" title="E133">A130</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E149:16"><span class="lab"><font color="Green" title="E127">A122</font></span></a>, <a class="txt" href="#E4:16_6_1"><span class="lab"><font color="Green" title="E132">A129</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E6:16_6_1"/>
( <font color="Maroon" title="c10">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> &amp; <font color="Maroon" title="c11">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E79:16"><span class="lab"><font color="Green" title="E85">A75</font></span></a>, <a class="txt" href="#E121:16"><span class="lab"><font color="Green" title="E111">A101</font></span></a>, <a class="txt" href="#E4:16_6_1"><span class="lab"><font color="Green" title="E132">A129</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E7:16_6_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c25">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_6_1"><span class="lab"><font color="Green" title="E133">A130</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:16_6_1"/>
 ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:16_6"/><span class="lab"><font color="Green" title="E132">A131</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> implies  ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_6_2"/><span class="lab"><font color="Green" title="E133">A132</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font></span><br/>

<a NAME="E2:16_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A133</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:16"><span class="lab"><font color="Green" title="E94">A84</font></span></a>, <a class="txt" href="#E151:16"><span class="lab"><font color="Green" title="E129">A126</font></span></a>, <a class="txt" href="#E101:16"><span class="lab"><font color="Green" title="E99">A89</font></span></a>, <a class="txt" href="#E95:16"><span class="lab"><font color="Green" title="E95">A85</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E3:16_6_2"/>
( <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E138:16"><span class="lab"><font color="Green" title="E120">A115</font></span></a>, <a class="txt" href="#E100:16"><span class="lab"><font color="Green" title="E98">A88</font></span></a>, <a class="txt" href="#E151:16"><span class="lab"><font color="Green" title="E129">A126</font></span></a>, <a class="txt" href="#E101:16"><span class="lab"><font color="Green" title="E99">A89</font></span></a>, <a class="txt" href="#E95:16"><span class="lab"><font color="Green" title="E95">A85</font></span></a>, <a class="txt" href="#E79:16"><span class="lab"><font color="Green" title="E85">A75</font></span></a>, <a class="txt" href="#E121:16"><span class="lab"><font color="Green" title="E111">A101</font></span></a>, <a class="txt" href="#E1:16_6_2"><span class="lab"><font color="Green" title="E133">A132</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/>
<a NAME="E4:16_6_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:16"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E2:16_6_2"><span class="lab"><font color="Green" title="E134">A133</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_6_2"/>
 ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:16_6"/>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font></span><br/>

<span class="kw">hence </span><a NAME="E4:16_6"/>
 ex <font color="Olive" title="b1">O</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E117:16"><span class="lab"><font color="Green" title="E109">A99</font></span></a>, <a class="txt" href="#E114:16"><span class="lab"><font color="Green" title="E106">A96</font></span></a>, <a class="txt" href="#E136:16"><span class="lab"><font color="Green" title="E119">A114</font></span></a>, <a class="txt" href="#E75:16"><span class="lab"><font color="Green" title="E83">A73</font></span></a>, <a class="txt" href="#E125:16"><span class="lab"><font color="Green" title="E112">A102</font></span></a>, <a class="txt" href="#E148:16"><span class="lab"><font color="Green" title="E126">A121</font></span></a>, <a class="txt" href="#E146:16"><span class="lab"><font color="Green" title="E125">A120</font></span></a>, <a class="txt" href="#E144:16"><span class="lab"><font color="Green" title="E124">A119</font></span></a>, <a class="txt" href="#E1:16_6"><span class="lab"><font color="Green" title="E131">A128</font></span></a>, <a class="txt" href="#E2:16_6"><span class="lab"><font color="Green" title="E132">A131</font></span></a>, <a class="ref" href="combgras.html#T7" target="_self" title="COMBGRAS:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E153:16"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="txt" href="#E127:16"><span class="lab"><font color="Green" title="E114">A104</font></span></a>, <a class="txt" href="#E125:16"><span class="lab"><font color="Green" title="E112">A102</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/>
<a NAME="E154:16"/><span class="kw">then </span>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E155:16"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive">b<sub>1</sub></font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E150:16"><span class="lab"><font color="Green" title="E128">A123</font></span></a>, <a class="txt" href="#E152:16"><span class="lab"><font color="Green" title="E130">A127</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
