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<b>let </b><font color="Maroon">G1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">G2</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> iff (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="supinf_2.html#K2">+</a> <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> ) )
 <b>and </b><br/><a NAME="E2:21_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> iff (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="supinf_2.html#K2">+</a> <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> ) )
 ;<br/>

<b>thus </b><a NAME="E3:21_1_2"/>
<font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G2</font>
  <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="21_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:21_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:21_1_2_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:21_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">y</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="E4:21_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E6:21_1_2"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:21_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 <b>by </b><i/>;<br/>


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