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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>thus </b><a NAME="E1:24"/>
(  not <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:24_1"/><b>then </b><i><font color="Green">E17</font></i>: 
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
;<br/>
<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E2:24_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="int_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="int_1.html#E1:24_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><a NAME="E2:24_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:24_1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="int_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E4:24_1_2"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E5:24_1_2"/><b>then </b>
(  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="txt" href="int_1.html#E1:24_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="int_1.html#E3:24_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/><b>hence </b><a NAME="E6:24_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:24_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) )
 <b>by </b><i><a class="txt" href="int_1.html#E1:24_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="int_1.html#E2:24_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="int_1.html#E3:24_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:24"/>
( ( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) ) implies <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>


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