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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[ <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b>$1 is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:3_1_1"/><i><font color="Green" title="E3">A</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">s</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font>
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D1" target="_self" title="XXREAL_2:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">s</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green" title="E4">W1</font></i>: 
for <font color="Olive" title="b1">r</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font>
 <b>and </b><br/><a NAME="E4:3_1_1"/><i><font color="Green" title="E5">W2</font></i>: 
for <font color="Olive" title="b1">r</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 <b>from </b><i><a class="ref" href="xxreal_1.html#S1" title="XXREAL_1:sch.1">XXREAL_1:sch 1</a>(<a class="txt" href="#E1:3_1_1"><i><font color="Green" title="E3">A</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c2">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">s</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<b>thus </b><a NAME="E5:3_1_1"/>
<font color="Maroon" title="c2">s</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:3_1_1"><i><font color="Green" title="E4">W1</font></i></a>, <a class="ref" href="xxreal_2.html#D1" target="_self" title="XXREAL_2:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font></span><br/>

<b>thus </b><a NAME="E6:3_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="#E4:3_1_1"><i><font color="Green" title="E5">W2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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