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<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="firebrick">::  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 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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> 2 <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/> for <font color="Olive" title="b2">F</font> being    <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of  <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>), <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>)  st <font color="Olive" title="b2">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  holds <br/> for <font color="Olive" title="b3">K</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  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="Olive" title="b1">X</font>)</span>)</span>  st <font color="Olive" title="b3">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  holds <br/>( <font color="Olive" title="b2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b3">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  &amp; <font color="Olive" title="b2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b3">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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> 2 <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  for <font color="Olive" title="b1">F</font> being    <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of  <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>), <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>)  st <font color="Olive" title="b1">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  holds <br/> for <font color="Olive" title="b2">K</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  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>  st <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  holds <br/>( <font color="Olive" title="b1">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  &amp; <font color="Olive" title="b1">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being    <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of  <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>), <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>)  st <font color="Olive" title="b1">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  holds <br/> for <font color="Olive" title="b2">K</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  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>  st <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  holds <br/>( <font color="Olive" title="b1">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  &amp; <font color="Olive" title="b1">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">F</font> be    <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of  <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>), <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>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  implies  for <font color="Olive" title="b1">K</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  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>  st <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  holds <br/>( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:38"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  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>  st <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  holds <br/>( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  )</span><br/>

<a NAME="E4:38"/><span class="lab"><font color="Green" title="E32">A4</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="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</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="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="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>;<br/>
<a NAME="E6:38"/>
2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E7:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A6</font></span>: 
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E8:38"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A8</font></span>: 
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">K</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  implies ( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E10:38"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  )</span><br/>

<a NAME="E11:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="combgras.html#T14" target="_self" title="COMBGRAS:th.14">Th14</a></span>;<br/>
<a NAME="E12:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>;<br/>
<a NAME="E13:38"/>
<font color="Maroon" title="c1">k</font> <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> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E14:38"/><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> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E15:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A12</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E16:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A13</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="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E17:38"/><span class="lab"><font color="Green" title="E42">A14</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="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E18:38"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E19:38"/><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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A16</font></span>: 
1 <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="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E21:38"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
 not <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_1"/>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E3:38_1"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B</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 &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> <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> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> ) </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="combgras.html#D5" target="_self" title="COMBGRAS:def.5">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:38_1"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
( <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:38"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E6:38_1"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<font color="Maroon" title="c5">B</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="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a></span>;<br/>
<a NAME="E7:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A21</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">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E5:38_1"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:38_1"/><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font>
 <span class="kw">and </span><br/><a NAME="E10:38_1"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:38_1"/><span class="lab"><font color="Green" title="E52">A24</font></span>: 
<font color="Maroon" title="c8">m</font> <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="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a></span>;<br/>
<a NAME="E13:38_1"/><span class="lab"><font color="Green" title="E53">A25</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">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E6:38_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E9:38_1"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="txt" href="#E10:38_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E14:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38"><span class="lab"><font color="Green" title="E41">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">B2</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="E16:38_1"/><span class="lab"><font color="Green" title="E54">A26</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B2</font>
 ;<br/>
<a NAME="E17:38_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="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</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="#E6:38_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E7:38_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="txt" href="#E9:38_1"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="txt" href="#E10:38_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">X3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:38_1"/><span class="lab"><font color="Green" title="E55">A27</font></span>: 
<font color="Maroon" title="c10">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font>
 <span class="kw">and </span><br/><a NAME="E20:38_1"/><span class="lab"><font color="Green" title="E56">A28</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E21:38_1"/><span class="lab"><font color="Green" title="E57">A29</font></span>: 
<font color="Maroon" title="c10">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">X1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:38_1"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<a NAME="E22:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A30</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">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X3</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E6:38_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E20:38_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E23:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X3</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38"><span class="lab"><font color="Green" title="E41">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">B3</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="E25:38_1"/><span class="lab"><font color="Green" title="E59">A31</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B3</font>
 ;<br/>
<a NAME="E26:38_1"/>
<font color="Maroon" title="c5">B</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="#E17:38"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">L2</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="E28:38_1"/><span class="lab"><font color="Green" title="E60">A32</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">L2</font>
 ;<br/>
<a NAME="E29:38_1"/>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="txt" href="#E7:38_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">B1</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="E31:38_1"/><span class="lab"><font color="Green" title="E61">A33</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B1</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E33:38_1"/><span class="lab"><font color="Green" title="E62">A34</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c14">S</font> <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">and </span><br/><a NAME="E34:38_1"/><span class="lab"><font color="Green" title="E63">A35</font></span>: 
<font color="Maroon" title="c4">K</font> <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> &amp; <font color="Maroon" title="c14">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">A1</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="E36:38_1"/><span class="lab"><font color="Green" title="E64">A36</font></span>: 
<font color="Maroon" title="c15">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c13">B1</font>
 ;<br/>
<a NAME="E37:38_1"/><span class="lab"><font color="Green" title="E65">A37</font></span>: 
<font color="Maroon" title="c10">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:38_1"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<a NAME="E38:38_1"/><span class="lab"><font color="Green" title="E66">A38</font></span>: 
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X3</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_1_1"/>
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> or <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X3</font> or <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38_1"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="txt" href="#E19:38_1"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="txt" href="#E37:38_1"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="ref" href="xboole_1.html#T38" title="XBOOLE_1:th.38">XBOOLE_1:38</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:38_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="txt" href="#E20:38_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c16">A3</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="E40:38_1"/><span class="lab"><font color="Green" title="E67">A39</font></span>: 
<font color="Maroon" title="c16">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">B3</font>
 ;<br/>
<a NAME="E41:38_1"/><span class="lab"><font color="Green" title="E68">A40</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E42:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E22:38_1"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E25:38_1"><span class="lab"><font color="Green" title="E59">A31</font></span></a></span>;<br/>
<a NAME="E43:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A41</font></span>: 
<font color="Maroon" title="c16">A3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:38_1"><span class="lab"><font color="Green" title="E67">A39</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E44:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A42</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="Maroon" title="c16">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A13</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> &amp; <font color="Maroon" title="c14">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A13</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:38_1"><span class="lab"><font color="Green" title="E63">A35</font></span></a></span>;<br/>
<a NAME="E45:38_1"/><span class="lab"><font color="Green" title="E71">A43</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E46:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E7:38_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="txt" href="#E31:38_1"><span class="lab"><font color="Green" title="E61">A33</font></span></a></span>;<br/>
<a NAME="E47:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A44</font></span>: 
<font color="Maroon" title="c15">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:38_1"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E48:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A45</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="Maroon" title="c15">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A11</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> &amp; <font color="Maroon" title="c14">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A11</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:38_1"><span class="lab"><font color="Green" title="E63">A35</font></span></a></span>;<br/>
<a NAME="E49:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A46</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">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
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">A2</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="E51:38_1"/><span class="lab"><font color="Green" title="E75">A47</font></span>: 
<font color="Maroon" title="c17">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">B2</font>
 ;<br/>
<a NAME="E52:38_1"/><span class="lab"><font color="Green" title="E76">A48</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E53:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E13:38_1"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="txt" href="#E16:38_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a></span>;<br/>
<a NAME="E54:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A49</font></span>: 
<font color="Maroon" title="c17">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:38_1"><span class="lab"><font color="Green" title="E75">A47</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">L3a</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="E56:38_1"/><span class="lab"><font color="Green" title="E78">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="c15">A1</font>,<font color="Maroon" title="c17">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">L3a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E47:38_1"><span class="lab"><font color="Green" title="E72">A44</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E57:38_1"/><span class="lab"><font color="Green" title="E79">A51</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:38_1"><span class="lab"><font color="Green" title="E73">A45</font></span></a></span>;<br/>
<a NAME="E58:38_1"/><span class="lab"><font color="Green" title="E80">A52</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>;<br/>
<a NAME="E59:38_1"/><span class="lab"><font color="Green" title="E81">A53</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="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">A3</font></span>)</span> <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="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">L1</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="E61:38_1"/><span class="lab"><font color="Green" title="E82">A54</font></span>: 
<font color="Maroon" title="c19">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c12">L2</font>
 ;<br/>
<a NAME="E62:38_1"/>
<font color="Maroon" title="c13">B1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E45:38_1"><span class="lab"><font color="Green" title="E71">A43</font></span></a>, <a class="txt" href="#E31:38_1"><span class="lab"><font color="Green" title="E61">A33</font></span></a>, <a class="txt" href="#E28:38_1"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E63:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A55</font></span>: 
<font color="Maroon" title="c15">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:38_1"><span class="lab"><font color="Green" title="E80">A52</font></span></a>, <a class="txt" href="#E36:38_1"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="txt" href="#E61:38_1"><span class="lab"><font color="Green" title="E82">A54</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E64:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A56</font></span>: 
<font color="Maroon" title="c15">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E65:38_1"/>
<font color="Maroon" title="c19">L1</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="E66:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A57</font></span>: 
 ex <font color="Olive" title="b1">l12</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="Maroon" title="c19">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">l12</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="#E17:38"><span class="lab"><font color="Green" title="E42">A14</font></span></a></span>;<br/>
<a NAME="E67:38_1"/>
<font color="Maroon" title="c11">B3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E41:38_1"><span class="lab"><font color="Green" title="E68">A40</font></span></a>, <a class="txt" href="#E25:38_1"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="txt" href="#E28:38_1"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E68:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A58</font></span>: 
<font color="Maroon" title="c16">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:38_1"><span class="lab"><font color="Green" title="E80">A52</font></span></a>, <a class="txt" href="#E40:38_1"><span class="lab"><font color="Green" title="E67">A39</font></span></a>, <a class="txt" href="#E61:38_1"><span class="lab"><font color="Green" title="E82">A54</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E69:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A59</font></span>: 
<font color="Maroon" title="c16">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E70:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:38_1"><span class="lab"><font color="Green" title="E84">A56</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E71:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A60</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="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</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="#E66:38_1"><span class="lab"><font color="Green" title="E85">A57</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E72:38_1"/><span class="lab"><font color="Green" title="E89">A61</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="Maroon" title="c17">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A12</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> &amp; <font color="Maroon" title="c14">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A12</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:38_1"><span class="lab"><font color="Green" title="E77">A49</font></span></a>, <a class="txt" href="#E34:38_1"><span class="lab"><font color="Green" title="E63">A35</font></span></a></span>;<br/>
<a NAME="E73:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A62</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c17">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="E74:38_1"/>
<font color="Maroon" title="c9">B2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E52:38_1"><span class="lab"><font color="Green" title="E76">A48</font></span></a>, <a class="txt" href="#E16:38_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E28:38_1"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E75:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A63</font></span>: 
<font color="Maroon" title="c17">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:38_1"><span class="lab"><font color="Green" title="E80">A52</font></span></a>, <a class="txt" href="#E51:38_1"><span class="lab"><font color="Green" title="E75">A47</font></span></a>, <a class="txt" href="#E61:38_1"><span class="lab"><font color="Green" title="E82">A54</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E76:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A64</font></span>: 
<font color="Maroon" title="c17">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E77:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:38_1"><span class="lab"><font color="Green" title="E84">A56</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E78:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A65</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="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A2</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="#E66:38_1"><span class="lab"><font color="Green" title="E85">A57</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E79:38_1"/><span class="lab"><font color="Green" title="E94">A66</font></span>: 
(  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp;  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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a>, <a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E80:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A67</font></span>: 
<font color="Maroon" title="c15">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:38_1"><span class="lab"><font color="Green" title="E66">A38</font></span></a>, <a class="txt" href="#E31:38_1"><span class="lab"><font color="Green" title="E61">A33</font></span></a>, <a class="txt" href="#E16:38_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E36:38_1"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="txt" href="#E51:38_1"><span class="lab"><font color="Green" title="E75">A47</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E81:38_1"/><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="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:38_1"><span class="lab"><font color="Green" title="E73">A45</font></span></a>, <a class="txt" href="#E72:38_1"><span class="lab"><font color="Green" title="E89">A61</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E82:38_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A2</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="#E78:38_1"><span class="lab"><font color="Green" title="E93">A65</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="E83:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A68</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="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E72:38_1"><span class="lab"><font color="Green" title="E89">A61</font></span></a>, <a class="txt" href="#E49:38_1"><span class="lab"><font color="Green" title="E74">A46</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E84:38_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c15">A1</font>,<font color="Maroon" title="c17">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="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:38_1"><span class="lab"><font color="Green" title="E83">A55</font></span></a>, <a class="txt" href="#E75:38_1"><span class="lab"><font color="Green" title="E91">A63</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E85:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A69</font></span>: 
<font color="Maroon" title="c19">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L3a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:38_1"><span class="lab"><font color="Green" title="E95">A67</font></span></a>, <a class="txt" href="#E56:38_1"><span class="lab"><font color="Green" title="E78">A50</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">consider </span><font color="Maroon" title="c20">L3b</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="E87:38_1"/><span class="lab"><font color="Green" title="E98">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="c17">A2</font>,<font color="Maroon" title="c16">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="c20">L3b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E54:38_1"><span class="lab"><font color="Green" title="E77">A49</font></span></a>, <a class="txt" href="#E43:38_1"><span class="lab"><font color="Green" title="E69">A41</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E88:38_1"/>
<font color="Maroon" title="c15">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:38_1"><span class="lab"><font color="Green" title="E94">A66</font></span></a>, <a class="txt" href="#E38:38_1"><span class="lab"><font color="Green" title="E66">A38</font></span></a>, <a class="txt" href="#E31:38_1"><span class="lab"><font color="Green" title="E61">A33</font></span></a>, <a class="txt" href="#E25:38_1"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="txt" href="#E36:38_1"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="txt" href="#E40:38_1"><span class="lab"><font color="Green" title="E67">A39</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E89:38_1"/><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="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:38_1"><span class="lab"><font color="Green" title="E73">A45</font></span></a>, <a class="txt" href="#E44:38_1"><span class="lab"><font color="Green" title="E70">A42</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E90:38_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</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="#E71:38_1"><span class="lab"><font color="Green" title="E88">A60</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="E91:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A71</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="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">A3</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E44:38_1"><span class="lab"><font color="Green" title="E70">A42</font></span></a>, <a class="txt" href="#E49:38_1"><span class="lab"><font color="Green" title="E74">A46</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E92:38_1"/>
<font color="Maroon" title="c16">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:38_1"><span class="lab"><font color="Green" title="E98">A70</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E93:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A72</font></span>: 
<font color="Maroon" title="c16">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E94:38_1"/>
<font color="Maroon" title="c17">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:38_1"><span class="lab"><font color="Green" title="E98">A70</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E95:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A73</font></span>: 
<font color="Maroon" title="c17">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E96:38_1"/>
<font color="Maroon" title="c20">L3b</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="E97:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A74</font></span>: 
 ex <font color="Olive" title="b1">l13b</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="Maroon" title="c20">L3b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l13b</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">l13b</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="#E17:38"><span class="lab"><font color="Green" title="E42">A14</font></span></a></span>;<br/>
<a NAME="E98:38_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E80:38_1"><span class="lab"><font color="Green" title="E95">A67</font></span></a>, <a class="txt" href="#E48:38_1"><span class="lab"><font color="Green" title="E73">A45</font></span></a>, <a class="txt" href="#E72:38_1"><span class="lab"><font color="Green" title="E89">A61</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E99:38_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38_1"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E100:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A75</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="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">A3</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:38_1"><span class="lab"><font color="Green" title="E81">A53</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E101:38_1"/>
<font color="Maroon" title="c14">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:38_1"><span class="lab"><font color="Green" title="E73">A45</font></span></a>, <a class="txt" href="#E72:38_1"><span class="lab"><font color="Green" title="E89">A61</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E102:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:38_1"><span class="lab"><font color="Green" title="E70">A42</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E103:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">m</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">A3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:38_1"><span class="lab"><font color="Green" title="E62">A34</font></span></a>, <a class="txt" href="#E12:38_1"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E104:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A76</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c15">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">A3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38_1"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="txt" href="#E100:38_1"><span class="lab"><font color="Green" title="E103">A75</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="E105:38_1"/>
<font color="Maroon" title="c15">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">L3a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:38_1"><span class="lab"><font color="Green" title="E78">A50</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E106:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A77</font></span>: 
<font color="Maroon" title="c15">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">L3a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E107:38_1"/><span class="lab"><font color="Green" title="E106">A78</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 3
 
;<br/>
<a NAME="E108:38_1"/>
<font color="Maroon" title="c17">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:38_1"><span class="lab"><font color="Green" title="E92">A64</font></span></a>, <a class="txt" href="#E69:38_1"><span class="lab"><font color="Green" title="E87">A59</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E109:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A79</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="c17">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</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="#E66:38_1"><span class="lab"><font color="Green" title="E85">A57</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E110:38_1"/><span class="lab"><font color="Green" title="E108">A80</font></span>: 
<font color="Maroon" title="c17">A2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:38_1"><span class="lab"><font color="Green" title="E94">A66</font></span></a>, <a class="txt" href="#E38:38_1"><span class="lab"><font color="Green" title="E66">A38</font></span></a>, <a class="txt" href="#E16:38_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E25:38_1"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="txt" href="#E51:38_1"><span class="lab"><font color="Green" title="E75">A47</font></span></a>, <a class="txt" href="#E40:38_1"><span class="lab"><font color="Green" title="E67">A39</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E111:38_1"/><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="c17">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:38_1"><span class="lab"><font color="Green" title="E89">A61</font></span></a>, <a class="txt" href="#E44:38_1"><span class="lab"><font color="Green" title="E70">A42</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E112:38_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</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="#E109:38_1"><span class="lab"><font color="Green" title="E107">A79</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="E113:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A81</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="c17">A2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">A3</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E44:38_1"><span class="lab"><font color="Green" title="E70">A42</font></span></a>, <a class="txt" href="#E73:38_1"><span class="lab"><font color="Green" title="E90">A62</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E114:38_1"/>
( 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> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E18:38"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E115:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A82</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="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="#E72:38_1"><span class="lab"><font color="Green" title="E89">A61</font></span></a>, <a class="txt" href="#E44:38_1"><span class="lab"><font color="Green" title="E70">A42</font></span></a>, <a class="txt" href="#E104:38_1"><span class="lab"><font color="Green" title="E104">A76</font></span></a>, <a class="txt" href="#E83:38_1"><span class="lab"><font color="Green" title="E96">A68</font></span></a>, <a class="txt" href="#E57:38_1"><span class="lab"><font color="Green" title="E79">A51</font></span></a>, <a class="txt" href="#E113:38_1"><span class="lab"><font color="Green" title="E109">A81</font></span></a>, <a class="txt" href="#E91:38_1"><span class="lab"><font color="Green" title="E99">A71</font></span></a>, <a class="txt" href="#E107:38_1"><span class="lab"><font color="Green" title="E106">A78</font></span></a>, <a class="ref" href="combgras.html#T7" target="_self" title="COMBGRAS:th.7">Th7</a></span>;<br/>
<a NAME="E116:38_1"/><span class="lab"><font color="Green" title="E111">A83</font></span>: 
<font color="Maroon" title="c18">L3a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">L3b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_1_2"/>
<font color="Maroon" title="c18">L3a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">L3b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:38_1"><span class="lab"><font color="Green" title="E105">A77</font></span></a>, <a class="txt" href="#E95:38_1"><span class="lab"><font color="Green" title="E101">A73</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E3:38_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E93:38_1"><span class="lab"><font color="Green" title="E100">A72</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E4:38_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="#E115:38_1"><span class="lab"><font color="Green" title="E110">A82</font></span></a>, <a class="txt" href="#E97:38_1"><span class="lab"><font color="Green" title="E102">A74</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E5:38_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E117:38_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c17">A2</font>,<font color="Maroon" title="c16">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="c19">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:38_1"><span class="lab"><font color="Green" title="E91">A63</font></span></a>, <a class="txt" href="#E68:38_1"><span class="lab"><font color="Green" title="E86">A58</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E118:38_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E110:38_1"><span class="lab"><font color="Green" title="E108">A80</font></span></a>, <a class="txt" href="#E87:38_1"><span class="lab"><font color="Green" title="E98">A70</font></span></a>, <a class="txt" href="#E116:38_1"><span class="lab"><font color="Green" title="E111">A83</font></span></a>, <a class="txt" href="#E85:38_1"><span class="lab"><font color="Green" title="E97">A69</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:38"/><span class="lab"><font color="Green" title="E46">A84</font></span>: 
 not <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:38_2"/><span class="lab"><font color="Green" title="E47">A85</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 3
 
;<br/>
<span class="kw">assume </span><a NAME="E2:38_2"/>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E4:38_2"/><span class="lab"><font color="Green" title="E48">A86</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B</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 &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> <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> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> ) </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="combgras.html#D5" target="_self" title="COMBGRAS:def.5">Def5</a></span>;<br/>
<a NAME="E5:38_2"/>
<font color="Maroon" title="c5">B</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="#E17:38"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">L2</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="E7:38_2"/><span class="lab"><font color="Green" title="E49">A87</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L2</font>
 ;<br/>
<a NAME="E8:38_2"/>
 the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>;<br/>
<a NAME="E9:38_2"/><span class="kw">then </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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a>  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">l1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:38_2"/><span class="lab"><font color="Green" title="E50">A88</font></span>: 
<font color="Maroon" title="c7">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E12:38_2"/><span class="lab"><font color="Green" title="E51">A89</font></span>: 
<font color="Maroon" title="c6">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">L1</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="E14:38_2"/><span class="lab"><font color="Green" title="E52">A90</font></span>: 
<font color="Maroon" title="c7">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:38_2"><span class="lab"><font color="Green" title="E50">A88</font></span></a></span>;<br/>
<a NAME="E15:38_2"/><span class="lab"><font color="Green" title="E53">A91</font></span>: 
<font color="Maroon" title="c6">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38_2"><span class="lab"><font color="Green" title="E51">A89</font></span></a>, <a class="txt" href="#E14:38_2"><span class="lab"><font color="Green" title="E52">A90</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:38_2"/><span class="lab"><font color="Green" title="E54">A92</font></span>: 
( <font color="Maroon" title="c9">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:38"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E18:38_2"/><span class="lab"><font color="Green" title="E55">A93</font></span>: 
<font color="Maroon" title="c5">B</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="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a></span>;<br/>
<a NAME="E19:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A94</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">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E17:38_2"><span class="lab"><font color="Green" title="E54">A92</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E21:38_2"/><span class="lab"><font color="Green" title="E57">A95</font></span>: 
<font color="Maroon" title="c10">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font>
 <span class="kw">and </span><br/><a NAME="E22:38_2"/><span class="lab"><font color="Green" title="E58">A96</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E24:38_2"/><span class="lab"><font color="Green" title="E59">A97</font></span>: 
<font color="Maroon" title="c11">m</font> <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="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a></span>;<br/>
<a NAME="E25:38_2"/>
 <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">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</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="#E18:38_2"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="txt" href="#E19:38_2"><span class="lab"><font color="Green" title="E56">A94</font></span></a>, <a class="txt" href="#E21:38_2"><span class="lab"><font color="Green" title="E57">A95</font></span></a>, <a class="txt" href="#E22:38_2"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">X3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E27:38_2"/><span class="lab"><font color="Green" title="E60">A98</font></span>: 
<font color="Maroon" title="c12">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</font>
 <span class="kw">and </span><br/><a NAME="E28:38_2"/><span class="lab"><font color="Green" title="E61">A99</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c12">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E29:38_2"/><span class="lab"><font color="Green" title="E62">A100</font></span>: 
<font color="Maroon" title="c12">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">X1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:38_2"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<a NAME="E30:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A101</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">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X3</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E18:38_2"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="txt" href="#E28:38_2"><span class="lab"><font color="Green" title="E61">A99</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E31:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X3</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38"><span class="lab"><font color="Green" title="E41">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">B3</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="E33:38_2"/><span class="lab"><font color="Green" title="E64">A102</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B3</font>
 ;<br/>
<a NAME="E34:38_2"/>
<font color="Maroon" title="c8">L1</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="E35:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A103</font></span>: 
 ex <font color="Olive" title="b1">l12</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="Maroon" title="c8">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">l12</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="#E17:38"><span class="lab"><font color="Green" title="E42">A14</font></span></a></span>;<br/>
<a NAME="E36:38_2"/>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="txt" href="#E19:38_2"><span class="lab"><font color="Green" title="E56">A94</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">B1</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="E38:38_2"/><span class="lab"><font color="Green" title="E66">A104</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B1</font>
 ;<br/>
<a NAME="E39:38_2"/><span class="lab"><font color="Green" title="E67">A105</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E40:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A106</font></span>: 
<font color="Maroon" title="c14">B1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E38:38_2"><span class="lab"><font color="Green" title="E66">A104</font></span></a>, <a class="txt" href="#E7:38_2"><span class="lab"><font color="Green" title="E49">A87</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E42:38_2"/><span class="lab"><font color="Green" title="E69">A107</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">S</font> <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">and </span><br/><a NAME="E43:38_2"/><span class="lab"><font color="Green" title="E70">A108</font></span>: 
<font color="Maroon" title="c4">K</font> <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> &amp; <font color="Maroon" title="c15">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></span>;<br/>
<a NAME="E44:38_2"/><span class="lab"><font color="Green" title="E71">A109</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>;<br/>
<a NAME="E45:38_2"/><span class="lab"><font color="Green" title="E72">A110</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E46:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A111</font></span>: 
<font color="Maroon" title="c13">B3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E33:38_2"><span class="lab"><font color="Green" title="E64">A102</font></span></a>, <a class="txt" href="#E7:38_2"><span class="lab"><font color="Green" title="E49">A87</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E47:38_2"/><span class="lab"><font color="Green" title="E74">A112</font></span>: 
 the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>;<br/>
<a NAME="E48:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A113</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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">a3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E50:38_2"/><span class="lab"><font color="Green" title="E76">A114</font></span>: 
<font color="Maroon" title="c16">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E51:38_2"/><span class="lab"><font color="Green" title="E77">A115</font></span>: 
<font color="Maroon" title="c13">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">A3</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="E53:38_2"/><span class="lab"><font color="Green" title="E78">A116</font></span>: 
<font color="Maroon" title="c16">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">A3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:38_2"><span class="lab"><font color="Green" title="E76">A114</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E55:38_2"/><span class="lab"><font color="Green" title="E79">A117</font></span>: 
<font color="Maroon" title="c18">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E56:38_2"/><span class="lab"><font color="Green" title="E80">A118</font></span>: 
<font color="Maroon" title="c14">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:38_2"><span class="lab"><font color="Green" title="E75">A113</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">A1</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="E58:38_2"/><span class="lab"><font color="Green" title="E81">A119</font></span>: 
<font color="Maroon" title="c18">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:38_2"><span class="lab"><font color="Green" title="E79">A117</font></span></a></span>;<br/>
<a NAME="E59:38_2"/>
<font color="Maroon" title="c13">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E30:38_2"><span class="lab"><font color="Green" title="E63">A101</font></span></a>, <a class="txt" href="#E45:38_2"><span class="lab"><font color="Green" title="E72">A110</font></span></a>, <a class="txt" href="#E33:38_2"><span class="lab"><font color="Green" title="E64">A102</font></span></a></span>;<br/>
<a NAME="E60:38_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">C3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">C3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c13">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E61:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A120</font></span>: 
<font color="Maroon" title="c17">A3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:38_2"><span class="lab"><font color="Green" title="E74">A112</font></span></a>, <a class="txt" href="#E50:38_2"><span class="lab"><font color="Green" title="E76">A114</font></span></a>, <a class="txt" href="#E51:38_2"><span class="lab"><font color="Green" title="E77">A115</font></span></a>, <a class="txt" href="#E53:38_2"><span class="lab"><font color="Green" title="E78">A116</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E62:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A121</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="Maroon" title="c17">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A13</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> &amp; <font color="Maroon" title="c15">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A13</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:38_2"><span class="lab"><font color="Green" title="E70">A108</font></span></a></span>;<br/>
<a NAME="E63:38_2"/>
<font color="Maroon" title="c14">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E19:38_2"><span class="lab"><font color="Green" title="E56">A94</font></span></a>, <a class="txt" href="#E39:38_2"><span class="lab"><font color="Green" title="E67">A105</font></span></a>, <a class="txt" href="#E38:38_2"><span class="lab"><font color="Green" title="E66">A104</font></span></a></span>;<br/>
<a NAME="E64:38_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c14">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E65:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A122</font></span>: 
<font color="Maroon" title="c19">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:38_2"><span class="lab"><font color="Green" title="E74">A112</font></span></a>, <a class="txt" href="#E55:38_2"><span class="lab"><font color="Green" title="E79">A117</font></span></a>, <a class="txt" href="#E56:38_2"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E58:38_2"><span class="lab"><font color="Green" title="E81">A119</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E66:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A123</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="Maroon" title="c19">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A11</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> &amp; <font color="Maroon" title="c15">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A11</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:38_2"><span class="lab"><font color="Green" title="E70">A108</font></span></a></span>;<br/>
<a NAME="E67:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A124</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c19">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
 
;<br/>
<a NAME="E68:38_2"/><span class="lab"><font color="Green" title="E87">A125</font></span>: 
<font color="Maroon" title="c14">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c19">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:38_2"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E58:38_2"><span class="lab"><font color="Green" title="E81">A119</font></span></a></span>;<br/>
<a NAME="E69:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:38_2"><span class="lab"><font color="Green" title="E71">A109</font></span></a>, <a class="txt" href="#E40:38_2"><span class="lab"><font color="Green" title="E68">A106</font></span></a>, <a class="txt" href="#E15:38_2"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E70:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A126</font></span>: 
<font color="Maroon" title="c19">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E71:38_2"/><span class="lab"><font color="Green" title="E89">A127</font></span>: 
<font color="Maroon" title="c13">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c17">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:38_2"><span class="lab"><font color="Green" title="E77">A115</font></span></a>, <a class="txt" href="#E53:38_2"><span class="lab"><font color="Green" title="E78">A116</font></span></a></span>;<br/>
<a NAME="E72:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:38_2"><span class="lab"><font color="Green" title="E71">A109</font></span></a>, <a class="txt" href="#E46:38_2"><span class="lab"><font color="Green" title="E73">A111</font></span></a>, <a class="txt" href="#E15:38_2"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E73:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A128</font></span>: 
<font color="Maroon" title="c17">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E74:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:38_2"><span class="lab"><font color="Green" title="E88">A126</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E75:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A129</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="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</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="#E35:38_2"><span class="lab"><font color="Green" title="E65">A103</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E76:38_2"/><span class="lab"><font color="Green" title="E92">A130</font></span>: 
<font color="Maroon" title="c12">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:38_2"><span class="lab"><font color="Green" title="E62">A100</font></span></a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<a NAME="E77:38_2"/><span class="lab"><font color="Green" title="E93">A131</font></span>: 
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X3</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_2_1"/>
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</font> or <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X3</font> or <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c12">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c12">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:38_2"><span class="lab"><font color="Green" title="E57">A95</font></span></a>, <a class="txt" href="#E27:38_2"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="txt" href="#E76:38_2"><span class="lab"><font color="Green" title="E92">A130</font></span></a>, <a class="ref" href="xboole_1.html#T38" title="XBOOLE_1:th.38">XBOOLE_1:38</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:38_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:38_2"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="txt" href="#E28:38_2"><span class="lab"><font color="Green" title="E61">A99</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E78:38_2"/><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="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:38_2"><span class="lab"><font color="Green" title="E66">A104</font></span></a>, <a class="txt" href="#E33:38_2"><span class="lab"><font color="Green" title="E64">A102</font></span></a>, <a class="txt" href="#E56:38_2"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E51:38_2"><span class="lab"><font color="Green" title="E77">A115</font></span></a>, <a class="txt" href="#E58:38_2"><span class="lab"><font color="Green" title="E81">A119</font></span></a>, <a class="txt" href="#E53:38_2"><span class="lab"><font color="Green" title="E78">A116</font></span></a>, <a class="txt" href="#E66:38_2"><span class="lab"><font color="Green" title="E85">A123</font></span></a>, <a class="txt" href="#E62:38_2"><span class="lab"><font color="Green" title="E83">A121</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E79:38_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</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="#E75:38_2"><span class="lab"><font color="Green" title="E91">A129</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="E80:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A132</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="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A3</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E62:38_2"><span class="lab"><font color="Green" title="E83">A121</font></span></a>, <a class="txt" href="#E67:38_2"><span class="lab"><font color="Green" title="E86">A124</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E81:38_2"/><span class="lab"><font color="Green" title="E95">A133</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">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E18:38_2"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="txt" href="#E21:38_2"><span class="lab"><font color="Green" title="E57">A95</font></span></a>, <a class="txt" href="#E22:38_2"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E82:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38"><span class="lab"><font color="Green" title="E41">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">B2</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="E84:38_2"/><span class="lab"><font color="Green" title="E96">A134</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">B2</font>
 ;<br/>
<a NAME="E85:38_2"/><span class="lab"><font color="Green" title="E97">A135</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></span>;<br/>
<a NAME="E86:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A136</font></span>: 
<font color="Maroon" title="c20">B2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E84:38_2"><span class="lab"><font color="Green" title="E96">A134</font></span></a>, <a class="txt" href="#E7:38_2"><span class="lab"><font color="Green" title="E49">A87</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E88:38_2"/><span class="lab"><font color="Green" title="E99">A137</font></span>: 
<font color="Maroon" title="c21">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E89:38_2"/><span class="lab"><font color="Green" title="E100">A138</font></span>: 
<font color="Maroon" title="c20">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:38_2"><span class="lab"><font color="Green" title="E75">A113</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">A2</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="E91:38_2"/><span class="lab"><font color="Green" title="E101">A139</font></span>: 
<font color="Maroon" title="c21">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:38_2"><span class="lab"><font color="Green" title="E99">A137</font></span></a></span>;<br/>
<a NAME="E92:38_2"/>
<font color="Maroon" title="c20">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E81:38_2"><span class="lab"><font color="Green" title="E95">A133</font></span></a>, <a class="txt" href="#E85:38_2"><span class="lab"><font color="Green" title="E97">A135</font></span></a>, <a class="txt" href="#E84:38_2"><span class="lab"><font color="Green" title="E96">A134</font></span></a></span>;<br/>
<a NAME="E93:38_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c20">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E94:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A140</font></span>: 
<font color="Maroon" title="c22">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:38_2"><span class="lab"><font color="Green" title="E74">A112</font></span></a>, <a class="txt" href="#E88:38_2"><span class="lab"><font color="Green" title="E99">A137</font></span></a>, <a class="txt" href="#E89:38_2"><span class="lab"><font color="Green" title="E100">A138</font></span></a>, <a class="txt" href="#E91:38_2"><span class="lab"><font color="Green" title="E101">A139</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E95:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A141</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="Maroon" title="c22">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A12</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> &amp; <font color="Maroon" title="c15">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A12</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:38_2"><span class="lab"><font color="Green" title="E70">A108</font></span></a></span>;<br/>
<a NAME="E96:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A142</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c22">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="E97:38_2"/><span class="lab"><font color="Green" title="E105">A143</font></span>: 
<font color="Maroon" title="c20">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c22">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:38_2"><span class="lab"><font color="Green" title="E100">A138</font></span></a>, <a class="txt" href="#E91:38_2"><span class="lab"><font color="Green" title="E101">A139</font></span></a></span>;<br/>
<a NAME="E98:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:38_2"><span class="lab"><font color="Green" title="E71">A109</font></span></a>, <a class="txt" href="#E86:38_2"><span class="lab"><font color="Green" title="E98">A136</font></span></a>, <a class="txt" href="#E15:38_2"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E99:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A144</font></span>: 
<font color="Maroon" title="c22">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E100:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c22">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:38_2"><span class="lab"><font color="Green" title="E88">A126</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E101:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A145</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="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c22">A2</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="#E35:38_2"><span class="lab"><font color="Green" title="E65">A103</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E102:38_2"/>
<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="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c22">A2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:38_2"><span class="lab"><font color="Green" title="E93">A131</font></span></a>, <a class="txt" href="#E38:38_2"><span class="lab"><font color="Green" title="E66">A104</font></span></a>, <a class="txt" href="#E84:38_2"><span class="lab"><font color="Green" title="E96">A134</font></span></a>, <a class="txt" href="#E56:38_2"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E89:38_2"><span class="lab"><font color="Green" title="E100">A138</font></span></a>, <a class="txt" href="#E58:38_2"><span class="lab"><font color="Green" title="E81">A119</font></span></a>, <a class="txt" href="#E91:38_2"><span class="lab"><font color="Green" title="E101">A139</font></span></a>, <a class="txt" href="#E66:38_2"><span class="lab"><font color="Green" title="E85">A123</font></span></a>, <a class="txt" href="#E95:38_2"><span class="lab"><font color="Green" title="E103">A141</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E103:38_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c22">A2</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="#E101:38_2"><span class="lab"><font color="Green" title="E107">A145</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="E104:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A146</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="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E95:38_2"><span class="lab"><font color="Green" title="E103">A141</font></span></a>, <a class="txt" href="#E67:38_2"><span class="lab"><font color="Green" title="E86">A124</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E105:38_2"/><span class="lab"><font color="Green" title="E109">A147</font></span>: 
<font color="Maroon" title="c22">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:38_2"><span class="lab"><font color="Green" title="E71">A109</font></span></a>, <a class="txt" href="#E86:38_2"><span class="lab"><font color="Green" title="E98">A136</font></span></a>, <a class="txt" href="#E97:38_2"><span class="lab"><font color="Green" title="E105">A143</font></span></a>, <a class="txt" href="#E15:38_2"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E106:38_2"/>
<font color="Maroon" title="c22">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E99:38_2"><span class="lab"><font color="Green" title="E106">A144</font></span></a>, <a class="txt" href="#E73:38_2"><span class="lab"><font color="Green" title="E90">A128</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E107:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A148</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="c22">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</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="#E35:38_2"><span class="lab"><font color="Green" title="E65">A103</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E108:38_2"/>
<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="c22">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:38_2"><span class="lab"><font color="Green" title="E93">A131</font></span></a>, <a class="txt" href="#E84:38_2"><span class="lab"><font color="Green" title="E96">A134</font></span></a>, <a class="txt" href="#E33:38_2"><span class="lab"><font color="Green" title="E64">A102</font></span></a>, <a class="txt" href="#E89:38_2"><span class="lab"><font color="Green" title="E100">A138</font></span></a>, <a class="txt" href="#E51:38_2"><span class="lab"><font color="Green" title="E77">A115</font></span></a>, <a class="txt" href="#E91:38_2"><span class="lab"><font color="Green" title="E101">A139</font></span></a>, <a class="txt" href="#E53:38_2"><span class="lab"><font color="Green" title="E78">A116</font></span></a>, <a class="txt" href="#E95:38_2"><span class="lab"><font color="Green" title="E103">A141</font></span></a>, <a class="txt" href="#E62:38_2"><span class="lab"><font color="Green" title="E83">A121</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E109:38_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</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="#E107:38_2"><span class="lab"><font color="Green" title="E110">A148</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="E110:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A149</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="c22">A2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A3</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E62:38_2"><span class="lab"><font color="Green" title="E83">A121</font></span></a>, <a class="txt" href="#E96:38_2"><span class="lab"><font color="Green" title="E104">A142</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E111:38_2"/><span class="lab"><font color="Green" title="E112">A150</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c19">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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:38_2"><span class="lab"><font color="Green" title="E85">A123</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">L3a</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="E113:38_2"/><span class="lab"><font color="Green" title="E113">A151</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c22">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="c23">L3a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E65:38_2"><span class="lab"><font color="Green" title="E84">A122</font></span></a>, <a class="txt" href="#E94:38_2"><span class="lab"><font color="Green" title="E102">A140</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E114:38_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:38_2"><span class="lab"><font color="Green" title="E93">A131</font></span></a>, <a class="txt" href="#E38:38_2"><span class="lab"><font color="Green" title="E66">A104</font></span></a>, <a class="txt" href="#E84:38_2"><span class="lab"><font color="Green" title="E96">A134</font></span></a>, <a class="txt" href="#E56:38_2"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E89:38_2"><span class="lab"><font color="Green" title="E100">A138</font></span></a>, <a class="txt" href="#E58:38_2"><span class="lab"><font color="Green" title="E81">A119</font></span></a>, <a class="txt" href="#E91:38_2"><span class="lab"><font color="Green" title="E101">A139</font></span></a>, <a class="txt" href="#E66:38_2"><span class="lab"><font color="Green" title="E85">A123</font></span></a>, <a class="txt" href="#E95:38_2"><span class="lab"><font color="Green" title="E103">A141</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E115:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A152</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="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E24:38_2"><span class="lab"><font color="Green" title="E59">A97</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E116:38_2"/>
 <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="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A3</font></span>)</span> <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="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E117:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A153</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="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A3</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:38_2"><span class="lab"><font color="Green" title="E114">A152</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E118:38_2"/>
<font color="Maroon" title="c15">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:38_2"><span class="lab"><font color="Green" title="E85">A123</font></span></a>, <a class="txt" href="#E95:38_2"><span class="lab"><font color="Green" title="E103">A141</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E119:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:38_2"><span class="lab"><font color="Green" title="E83">A121</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E120:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">m</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:38_2"><span class="lab"><font color="Green" title="E69">A107</font></span></a>, <a class="txt" href="#E24:38_2"><span class="lab"><font color="Green" title="E59">A97</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E121:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A154</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c19">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">A3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:38_2"><span class="lab"><font color="Green" title="E59">A97</font></span></a>, <a class="txt" href="#E117:38_2"><span class="lab"><font color="Green" title="E115">A153</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="E122:38_2"/>
<font color="Maroon" title="c19">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">L3a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E113:38_2"><span class="lab"><font color="Green" title="E113">A151</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E123:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A155</font></span>: 
<font color="Maroon" title="c19">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">L3a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">L3b</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="E125:38_2"/><span class="lab"><font color="Green" title="E118">A156</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c22">A2</font>,<font color="Maroon" title="c17">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="c24">L3b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E94:38_2"><span class="lab"><font color="Green" title="E102">A140</font></span></a>, <a class="txt" href="#E61:38_2"><span class="lab"><font color="Green" title="E82">A120</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E126:38_2"/>
<font color="Maroon" title="c17">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E125:38_2"><span class="lab"><font color="Green" title="E118">A156</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E127:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A157</font></span>: 
<font color="Maroon" title="c17">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E128:38_2"/>
<font color="Maroon" title="c22">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E125:38_2"><span class="lab"><font color="Green" title="E118">A156</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E129:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A158</font></span>: 
<font color="Maroon" title="c22">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E130:38_2"/>
<font color="Maroon" title="c24">L3b</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="E131:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A159</font></span>: 
 ex <font color="Olive" title="b1">l13b</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="Maroon" title="c24">L3b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l13b</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">l13b</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="#E17:38"><span class="lab"><font color="Green" title="E42">A14</font></span></a></span>;<br/>
<a NAME="E132:38_2"/>
( 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> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E18:38"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E133:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A160</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="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="#E95:38_2"><span class="lab"><font color="Green" title="E103">A141</font></span></a>, <a class="txt" href="#E62:38_2"><span class="lab"><font color="Green" title="E83">A121</font></span></a>, <a class="txt" href="#E121:38_2"><span class="lab"><font color="Green" title="E116">A154</font></span></a>, <a class="txt" href="#E104:38_2"><span class="lab"><font color="Green" title="E108">A146</font></span></a>, <a class="txt" href="#E111:38_2"><span class="lab"><font color="Green" title="E112">A150</font></span></a>, <a class="txt" href="#E110:38_2"><span class="lab"><font color="Green" title="E111">A149</font></span></a>, <a class="txt" href="#E80:38_2"><span class="lab"><font color="Green" title="E94">A132</font></span></a>, <a class="txt" href="#E1:38_2"><span class="lab"><font color="Green" title="E47">A85</font></span></a>, <a class="ref" href="combgras.html#T7" target="_self" title="COMBGRAS:th.7">Th7</a></span>;<br/>
<a NAME="E134:38_2"/><span class="lab"><font color="Green" title="E123">A161</font></span>: 
<font color="Maroon" title="c23">L3a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">L3b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_2_2"/>
<font color="Maroon" title="c23">L3a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">L3b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c22">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E123:38_2"><span class="lab"><font color="Green" title="E117">A155</font></span></a>, <a class="txt" href="#E129:38_2"><span class="lab"><font color="Green" title="E120">A158</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E3:38_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c22">A2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">L3b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E127:38_2"><span class="lab"><font color="Green" title="E119">A157</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E4:38_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="#E133:38_2"><span class="lab"><font color="Green" title="E122">A160</font></span></a>, <a class="txt" href="#E131:38_2"><span class="lab"><font color="Green" title="E121">A159</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E5:38_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E135:38_2"/>
<font color="Maroon" title="c19">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:38_2"><span class="lab"><font color="Green" title="E71">A109</font></span></a>, <a class="txt" href="#E40:38_2"><span class="lab"><font color="Green" title="E68">A106</font></span></a>, <a class="txt" href="#E68:38_2"><span class="lab"><font color="Green" title="E87">A125</font></span></a>, <a class="txt" href="#E15:38_2"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E136:38_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="c19">A1</font>,<font color="Maroon" title="c22">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="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:38_2"><span class="lab"><font color="Green" title="E109">A147</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E137:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A162</font></span>: 
<font color="Maroon" title="c8">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">L3a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:38_2"><span class="lab"><font color="Green" title="E93">A131</font></span></a>, <a class="txt" href="#E38:38_2"><span class="lab"><font color="Green" title="E66">A104</font></span></a>, <a class="txt" href="#E84:38_2"><span class="lab"><font color="Green" title="E96">A134</font></span></a>, <a class="txt" href="#E56:38_2"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E89:38_2"><span class="lab"><font color="Green" title="E100">A138</font></span></a>, <a class="txt" href="#E58:38_2"><span class="lab"><font color="Green" title="E81">A119</font></span></a>, <a class="txt" href="#E91:38_2"><span class="lab"><font color="Green" title="E101">A139</font></span></a>, <a class="txt" href="#E113:38_2"><span class="lab"><font color="Green" title="E113">A151</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="E138:38_2"/>
<font color="Maroon" title="c17">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:38_2"><span class="lab"><font color="Green" title="E71">A109</font></span></a>, <a class="txt" href="#E46:38_2"><span class="lab"><font color="Green" title="E73">A111</font></span></a>, <a class="txt" href="#E71:38_2"><span class="lab"><font color="Green" title="E89">A127</font></span></a>, <a class="txt" href="#E15:38_2"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E139:38_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="c22">A2</font>,<font color="Maroon" title="c17">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="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:38_2"><span class="lab"><font color="Green" title="E109">A147</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E140:38_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:38_2"><span class="lab"><font color="Green" title="E93">A131</font></span></a>, <a class="txt" href="#E84:38_2"><span class="lab"><font color="Green" title="E96">A134</font></span></a>, <a class="txt" href="#E33:38_2"><span class="lab"><font color="Green" title="E64">A102</font></span></a>, <a class="txt" href="#E89:38_2"><span class="lab"><font color="Green" title="E100">A138</font></span></a>, <a class="txt" href="#E51:38_2"><span class="lab"><font color="Green" title="E77">A115</font></span></a>, <a class="txt" href="#E91:38_2"><span class="lab"><font color="Green" title="E101">A139</font></span></a>, <a class="txt" href="#E53:38_2"><span class="lab"><font color="Green" title="E78">A116</font></span></a>, <a class="txt" href="#E125:38_2"><span class="lab"><font color="Green" title="E118">A156</font></span></a>, <a class="txt" href="#E134:38_2"><span class="lab"><font color="Green" title="E123">A161</font></span></a>, <a class="txt" href="#E137:38_2"><span class="lab"><font color="Green" title="E124">A162</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:38"/>
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E11:38"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="combgras.html#T22" target="_self" title="COMBGRAS:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:38"/>
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E22:38"><span class="lab"><font color="Green" title="E46">A84</font></span></a>, <a class="txt" href="#E21:38"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="combgras.html#T15" target="_self" title="COMBGRAS:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
