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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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="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> (<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>,<font color="Olive" title="b1">X</font>), <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</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>,<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">H</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  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b3">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b2">F</font> holds <br/> for <font color="Olive" title="b4">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Olive" title="b1">X</font>  st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b3">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b3">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b4">f</font>) holds <br/> <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b2">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Olive" title="b4">f</font>)</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"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="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> (<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>,<font color="Maroon" title="c2">X</font>), <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</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>,<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">H</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  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F</font> holds <br/> for <font color="Olive" title="b3">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font>  st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b2">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b3">f</font>) holds <br/> <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Olive" title="b3">f</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="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> (<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>,<font color="Maroon" title="c2">X</font>), <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</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>,<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">H</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  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F</font> holds <br/> for <font color="Olive" title="b3">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font>  st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b2">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b3">f</font>) holds <br/> <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Olive" title="b3">f</font>)</span><br/>

<a NAME="E3:48"/>
<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> 3
 
<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="E4:48"/><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> 3
 
<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="E5:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</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="#E2:48"><span class="lab"><font color="Green" title="E39">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="E6:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A4</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="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E7:48"/>
<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> 3
 
<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="E8:48"/><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> 3
 
<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="E9:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<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 <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="#E2:48"><span class="lab"><font color="Green" title="E39">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="E10:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A6</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> (<span class="p2">(<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>,<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> <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="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E11:48"/>
<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="E12:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c1">k</font> <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="E13:48"/>
<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="E14:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A8</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> <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/>
<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> (<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>,<font color="Maroon" title="c2">X</font>), <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</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>,<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">H</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  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> holds <br/> for <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font>  st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">f</font>) holds <br/> <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Olive" title="b2">f</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E15:48"/><span class="lab"><font color="Green" title="E46">A9</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">H</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  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> holds <br/> for <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font>  st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">f</font>) holds <br/> <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Olive" title="b2">f</font>)</span><br/>

<a NAME="E16:48"/><span class="lab"><font color="Green" title="E47">A10</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="#E15:48"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">H</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"> (  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> implies  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font>  st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c4">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">f</font>) holds <br/> <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Olive" title="b1">f</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E17:48"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
 the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</font> <a href="funct_2.html#R1" title="FUNCT_2: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 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="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font>  st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c4">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">f</font>) holds <br/> <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Olive" title="b1">f</font>)</span><br/>

<a NAME="E18:48"/><span class="lab"><font color="Green" title="E49">A12</font></span>: 
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <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"> (  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c4">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c5">f</font>) implies  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Maroon" title="c5">f</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E19:48"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
 <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c4">H</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c5">f</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Maroon" title="c5">f</font>)</span><br/>

<a NAME="E20:48"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> holds <br/> 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">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <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"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> implies  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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">A</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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:48_1"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">A1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:48_1"/><span class="lab"><font color="Green" title="E53">A16</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a></span>;<br/>
<a NAME="E6:48_1"/>
<span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c8">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<a NAME="E7:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:48"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E19:48"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:48_1"/>
 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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E52">A15</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="E21:48"/><span class="lab"><font color="Green" title="E52">A17</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> (<span class="p2">(<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>,<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> <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> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E22:48"/><span class="lab"><font color="Green" title="E53">A18</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="#E15:48"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>;<br/>
<a NAME="E23:48"/><span class="lab"><font color="Green" title="E54">A19</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> holds <br/> 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="Olive" title="b1">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <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"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> implies  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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">A</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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:48_2"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 ;<br/>
<a NAME="E4:48_2"/>
<font color="Maroon" title="c6">x</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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E55">A20</font></span></a></span>;<br/>
<a NAME="E5:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A21</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="c6">x</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> <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="#E21:48"><span class="lab"><font color="Green" title="E52">A17</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">B1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:48_2"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
<font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E8:48_2"/><span class="lab"><font color="Green" title="E58">A23</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">B1</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="#E14:48"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E9:48_2"/><span class="lab"><font color="Green" title="E59">A24</font></span>: 
<font color="Maroon" title="c6">x</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="#E5:48_2"><span class="lab"><font color="Green" title="E56">A21</font></span></a></span>;<br/>
<a NAME="E10:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">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="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E8:48_2"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>;<br/>
<a NAME="E11:48_2"/>
<font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E8:48_2"><span class="lab"><font color="Green" title="E58">A23</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E14:48_2"/><span class="lab"><font color="Green" title="E61">A26</font></span>: 
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B1</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">C1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:48_2"/><span class="lab"><font color="Green" title="E62">A27</font></span>: 
<font color="Maroon" title="c10">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B1</font>
 <span class="kw">and </span><br/><a NAME="E17:48_2"/><span class="lab"><font color="Green" title="E63">A28</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E8:48_2"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E18:48_2"/>
<font color="Maroon" title="c8">B1</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="#E8:48_2"><span class="lab"><font color="Green" title="E58">A23</font></span></a></span>;<br/>
<a NAME="E19:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A29</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="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_2"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="txt" href="#E17:48_2"><span class="lab"><font color="Green" title="E63">A28</font></span></a>, <a class="txt" href="#E9:48_2"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="txt" href="#E10:48_2"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a>, <a class="ref" href="xboole_1.html#T85" title="XBOOLE_1:th.85">XBOOLE_1:85</a></span>;<br/>
<a NAME="E20:48_2"/>
<font color="Maroon" title="c10">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E16:48_2"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E21:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A30</font></span>: 
<font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E22:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E23:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font></span>)</span> <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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E19:48_2"><span class="lab"><font color="Green" title="E64">A29</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E25:48_2"/><span class="lab"><font color="Green" title="E66">A31</font></span>: 
<font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font></span>)</span>
 ;<br/>
<a NAME="E26:48_2"/>
<font color="Maroon" title="c11">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E21:48_2"><span class="lab"><font color="Green" title="E65">A30</font></span></a>, <a class="txt" href="#E25:48_2"><span class="lab"><font color="Green" title="E66">A31</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E27:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E28:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A32</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E29:48_2"/>
<font color="Maroon" title="c8">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E21:48_2"><span class="lab"><font color="Green" title="E65">A30</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E30:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A33</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">b2</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E14:48_2"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="txt" href="#E25:48_2"><span class="lab"><font color="Green" title="E66">A31</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E31:48_2"/>
<font color="Maroon" title="c8">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E32:48_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"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E33:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A34</font></span>: 
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_2"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="txt" href="#E14:48_2"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="txt" href="#E25:48_2"><span class="lab"><font color="Green" title="E66">A31</font></span></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E34:48_2"/><span class="kw">then </span>
<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 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_2"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="txt" href="#E19:48_2"><span class="lab"><font color="Green" title="E64">A29</font></span></a>, <a class="txt" href="#E14:48_2"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="txt" href="#E25:48_2"><span class="lab"><font color="Green" title="E66">A31</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E35:48_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="c9">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:48_2"><span class="lab"><font color="Green" title="E68">A33</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="E36:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A35</font></span>: 
<font color="Maroon" title="c9">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E9:48_2"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="txt" href="#E21:48_2"><span class="lab"><font color="Green" title="E65">A30</font></span></a>, <a class="txt" href="#E14:48_2"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="txt" href="#E25:48_2"><span class="lab"><font color="Green" title="E66">A31</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E37:48_2"/>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
;<br/>
<a NAME="E38:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A36</font></span>: 
 ex <font color="Olive" title="b1">B12</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <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="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a></span>;<br/>
<a NAME="E39:48_2"/>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
;<br/>
<a NAME="E40:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A37</font></span>: 
 ex <font color="Olive" title="b1">B11</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B11</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <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="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a></span>;<br/>
<a NAME="E41:48_2"/>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
;<br/>
<a NAME="E42:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A38</font></span>: 
 ex <font color="Olive" title="b1">L1</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="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L1</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48"><span class="lab"><font color="Green" title="E52">A17</font></span></a></span>;<br/>
<a NAME="E43:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A39</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E44:48_2"/>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:48"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="txt" href="#E18:48"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E33:48_2"><span class="lab"><font color="Green" title="E69">A34</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="E45:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A40</font></span>: 
<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 <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="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:48_2"><span class="lab"><font color="Green" title="E72">A37</font></span></a>, <a class="txt" href="#E38:48_2"><span class="lab"><font color="Green" title="E71">A36</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E46:48_2"/>
<font color="Maroon" title="c9">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E14:48_2"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E47:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>;<br/>
<a NAME="E48:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A41</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E49:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:48_2"><span class="lab"><font color="Green" title="E67">A32</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E50:48_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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font></span>)</span></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> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:48_2"><span class="lab"><font color="Green" title="E73">A38</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E51:48_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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:48_2"><span class="lab"><font color="Green" title="E75">A40</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="E52:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A42</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font></span>)</span> <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="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:48_2"><span class="lab"><font color="Green" title="E76">A41</font></span></a>, <a class="txt" href="#E28:48_2"><span class="lab"><font color="Green" title="E67">A32</font></span></a>, <a class="txt" href="#E42:48_2"><span class="lab"><font color="Green" title="E73">A38</font></span></a>, <a class="txt" href="#E43:48_2"><span class="lab"><font color="Green" title="E74">A39</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E53:48_2"/><span class="lab"><font color="Green" title="E78">A43</font></span>: 
<span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<a NAME="E54:48_2"/><span class="lab"><font color="Green" title="E79">A44</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">b2</font></span>)</span> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="relat_1.html#T120" title="RELAT_1:th.120">RELAT_1:120</a></span>;<br/>
<a NAME="E55:48_2"/>
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font> &amp; <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c9">b1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E18:48"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E56:48_2"/>
 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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E36:48_2"><span class="lab"><font color="Green" title="E70">A35</font></span></a>, <a class="txt" href="#E52:48_2"><span class="lab"><font color="Green" title="E77">A42</font></span></a>, <a class="txt" href="#E53:48_2"><span class="lab"><font color="Green" title="E78">A43</font></span></a>, <a class="txt" href="#E54:48_2"><span class="lab"><font color="Green" title="E79">A44</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</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="E24:48"/><span class="lab"><font color="Green" title="E55">A45</font></span>: 
(  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span> &amp;  <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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E25:48"/>
 <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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> (<span class="p2">(<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>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E26:48"/><span class="kw">then </span>
 the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">f</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="funct_1.html#D11" title="FUNCT_1:def.11">FUNCT_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:48"/>
 <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</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>,<font color="Maroon" title="c5">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48"><span class="lab"><font color="Green" title="E55">A45</font></span></a>, <a class="txt" href="#E23:48"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="ref" href="funct_1.html#D11" title="FUNCT_1:def.11">FUNCT_1:def 11</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/>


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