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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:54"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E3:54"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E4:54"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T53" target="_self">Th53</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:54_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59" target="_self">Th59</a></i>;<br/><b>hence </b><a NAME="E3:54_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_2.html#E1:54"><i><font color="Green">E39</font></i></a>, <a class="txt" href="card_2.html#E3:54"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:54"/><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_2.html#E4:54"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:54"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_2.html#E4:54"><i><font color="Green">E41</font></i></a>, <a class="txt" href="card_2.html#E5:54"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:54"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_2.html#E6:54"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>hence </b><a NAME="E9:54"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="card_2.html#T58" target="_self">Th58</a></i>;<br/>


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