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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:14_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:14_1_2"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:14_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E3:14_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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