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<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c2">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">z</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">z</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="txt" href="int_4.html#E3:5_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>take </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">z</font>;<br/><b>thus </b><a NAME="E5:5_1"/>
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">z</font> )
 <b>by </b><i><a class="txt" href="int_4.html#E3:5_1"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c4">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><a NAME="E3:5"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c1">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:5"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c5">y</font>
;<br/>

<b>thus </b><a NAME="E1:5_2"/>
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="int_4.html#E3:5"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:5"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></i>;<br/>


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