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<b>let </b><font color="Maroon" title="c2">Y1</font>, <font color="Maroon" title="c3">Y2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_2"/><i><font color="Green" title="E20">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font> iff <font color="Olive" title="b1">x</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font> )
 <b>and </b><br/><a NAME="E2:31_1_2"/><i><font color="Green" title="E21">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y2</font> iff <font color="Olive" title="b1">x</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font> )
 ;<br/>

<a NAME="E3:31_1_2"/><i><font color="Green" title="E22">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<a NAME="E1:31_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font> iff <font color="Maroon" title="c4">x</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:31_1_2"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:31_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y2</font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E2:31_1_2"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:31_1_2"/>
<font color="Maroon" title="c2">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_2_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y2</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:31_1_2"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:31_1_2_2"/>
<font color="Maroon" title="c2">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


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

<a NAME="E5:31_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y2</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:31_1_2"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:31_1_2"/>
<font color="Maroon" title="c3">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


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