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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>thus </b><a NAME="E1:52"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> implies 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:52_1"/>
not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:52_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T39" target="_self">Th39</a></i>;<br/>
<b>hence </b><a NAME="E4:52_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_1.html#E1:52_1"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:52"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E3:52"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T41" target="_self">Th41</a></i>;<br/>


</div>
