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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>





<a NAME="E1:8"/><span class="lab"><font color="Green" title="E1">A1</font></span>: 
( <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="8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:8_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E2">A2</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D3" title="BVFUNC_1:def.3">BVFUNC_1:def 3</a></span>;<br/>
<a NAME="E3:8_1"/>
<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/>
<span class="kw">hence </span><a NAME="E4:8_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="ref" href="bvfunc_1.html#D3" title="BVFUNC_1:def.3">BVFUNC_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:8"/>
( <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="8_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:8_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E2">A3</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D3" title="BVFUNC_1:def.3">BVFUNC_1:def 3</a></span>;<br/>
<a NAME="E3:8_2"/>
<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/>
<span class="kw">hence </span><a NAME="E4:8_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_2"><span class="lab"><font color="Green" title="E2">A3</font></span></a>, <a class="ref" href="bvfunc_1.html#D3" title="BVFUNC_1:def.3">BVFUNC_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:8"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E1">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
