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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">X</font> holds <br/>for <font color="Olive" title="b2">A</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 holds <br/> <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">k</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> implies for <font color="Olive" title="b1">A</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 holds <br/> <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">k</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> )</span><br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E35">A1</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 holds <br/> <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">k</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a></span><br/>

<b>let </b><font color="Maroon" title="c3">A</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 implies  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">k</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> )</span><br/>

<b>assume </b><a NAME="E2:45"/><i><font color="Green" title="E36">A2</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">k</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a></span><br/>

<a NAME="E3:45"/>
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K6" title="COMBGRAS:func.6">^^</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E2:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="combgras.html#D13" target="_self" title="COMBGRAS:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E4:45"/>
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">k</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 <b>by </b><i><a class="txt" href="#E2:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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