<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><a NAME="E1:7"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 ;<br/>

<a NAME="E2:7"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">d</font> holds <br/><font color="Olive">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>assume </b><a NAME="E2:7_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E3:7_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:7_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E2:7_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_1_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:7_1_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>


</div>
