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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be  <a href="xboolean.html#V1" title="XBOOLEAN:attr.1">boolean</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<a NAME="E1:5"/><i><font color="Green" title="E1">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c2">y</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font> implies <font color="Maroon" title="c1">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c3">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" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c2">y</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b><i><font color="Green" title="E2">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D3" title="BVFUNC_1:def.3">BVFUNC_1:def 3</a></i>;<br/>
<a NAME="E3:5_1"/>
<font color="Maroon" title="c1">x</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font>
 
;<br/>
<b>hence </b><a NAME="E4:5_1"/>
<font color="Maroon" title="c1">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="binari_5.html#E2:5_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" title="BVFUNC_1:def.3">BVFUNC_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:5"/>
( <font color="Maroon" title="c1">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c3">z</font> implies <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c2">y</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">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" title="c1">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c3">z</font>
 ;<br/>

<a NAME="E2:5_2"/><b>then </b><i><font color="Green" title="E2">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D3" title="BVFUNC_1:def.3">BVFUNC_1:def 3</a></i>;<br/>
<a NAME="E3:5_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:5_2"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c2">y</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="binari_5.html#E2:5_2"><i><font color="Green" title="E2">A3</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" title="BVFUNC_1:def.3">BVFUNC_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c2">y</font> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <font color="Maroon" title="c3">z</font> iff <font color="Maroon" title="c1">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <font color="Maroon" title="c3">z</font> )
 <b>by </b><i><a class="txt" href="binari_5.html#E1:5"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>


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