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

<b>let </b><font color="Maroon">x</font> be  <a href="xboolean.html#V1">boolean</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be  <a href="xboolean.html#V1">boolean</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be  <a href="xboolean.html#V1">boolean</a>  <a href="hidden.html#M1">set</a> ;<br/>





<a NAME="E1:5"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">y</font> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font> implies <font color="Maroon">x</font> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">y</font> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">x</font> <a href="xboolean.html#K6">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D3">BVFUNC_1:def 3</a></i>;<br/>
<a NAME="E3:5_1"/>
<font color="Maroon">x</font> <a href="xboolean.html#K6">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">y</font></span>)</span> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font>
 
;<br/>
<b>hence </b><a NAME="E4:5_1"/>
<font color="Maroon">x</font> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D3">BVFUNC_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:5"/>
( <font color="Maroon">x</font> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">z</font> implies <font color="Maroon">x</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">y</font> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_2"/>
<font color="Maroon">x</font> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:5_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">y</font></span>)</span> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D3">BVFUNC_1:def 3</a></i>;<br/>
<a NAME="E3:5_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">y</font></span>)</span> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xboolean.html#K6">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:5_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">y</font> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D3">BVFUNC_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">y</font> <a href="xboolean.html#K6">=&gt;</a> <font color="Maroon">z</font> iff <font color="Maroon">x</font> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>


</div>
