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



<b>thus </b><a NAME="E1:7"/>
<span class="p1">(<span class="default"><a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<a NAME="E1:7_1"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E2:7_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</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="E3:7_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E4:7_1"/>
( <font color="Maroon">x</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">b</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:7_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:7_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:7"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</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="E4:7"/>
( 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:7"/><b>then </b>
( 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; ( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 ) )
 
<b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<a NAME="E6:7"/><b>then </b>
( <font color="Maroon">x</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">b</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" target="_self">seq</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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