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<b>let </b><font color="Maroon">b1</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b2</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>thus </b><a NAME="E1:6"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> implies  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:6_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E4:6_1"/><i><font color="Green">E49</font></i>: 
( 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:6_1"/>
<font color="Maroon">b1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:6_1"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:6_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:6_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b2</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:6"/><i><font color="Green">E50</font></i>: 
 <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2"/>
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:6_2"/><b>then </b>
<font color="Maroon">b1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b1</font>
 <b>by </b><i/>;<br/><a NAME="E3:6_2"/><b>then </b>
<font color="Maroon">b1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:6_2"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font>
 ;<br/>


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