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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="margrel1.html#V2">boolean</a>  <a href="hidden.html#M1">set</a> ;<br/>





<a NAME="E1:19"/><i><font color="Green">E1</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></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/>
<font color="Maroon">c<sub>1</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binarith.html#T20">BINARITH:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binarith.html#T9">BINARITH:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>
<b>hence </b><a NAME="E4:19_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="binari_5.html#E2:19_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="bvfunc_1.html#D3">BVFUNC_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:19"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:19_2"/><b>then </b><i><font color="Green">E2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></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/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binarith.html#T9">BINARITH:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T20">BINARITH:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E4:19_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="binari_5.html#E2:19_2"><i><font color="Green">E2</font></i></a>, <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="E3:19"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K1">'imp'</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="binari_5.html#E1:19"><i><font color="Green">E1</font></i></a></i>;<br/>


</div>
