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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">z</font> be   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:81"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">and </span><br/><a NAME="E2:81"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:81_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E52">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><span class="kw">suppose </span></a><a NAME="E1:81_1_1"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> )</span><br/><div class="add"><a NAME="E2:81_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="xxreal_3.html#T46" target="_self" title="XXREAL_3:th.46">Th50</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">y</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:81_1_1"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">and </span><br/><a NAME="E5:81_1_1"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T3" target="_self" title="XXREAL_3:th.3">Th3</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> )</span><br/><a NAME="E6:81_1_1"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="xxreal_3.html#T36" target="_self" title="XXREAL_3:th.36">Th38</a></span>;<br/><a NAME="E7:81_1_1"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T48" title="XXREAL_0:th.48">XXREAL_0:48</a></span>;<br/><a NAME="E8:81_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T22" target="_self" title="XXREAL_3:th.22">Th22</a></span>;<br/><a NAME="E9:81_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81_1_1"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E5:81_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="xxreal_3.html#T22" target="_self" title="XXREAL_3:th.22">Th22</a></span>;<br/><span class="kw">hence </span><a NAME="E10:81_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E6:81_1_1"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E8:81_1_1"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2"><span class="kw">suppose </span></a><a NAME="E1:81_1_2"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c3">y</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:81_1_2"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="xxreal_3.html#T3" target="_self" title="XXREAL_3:th.3">Th3</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> )</span><br/><span class="kw">thus </span><a NAME="E4:81_1_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_2"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E3:81_1_2"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="xxreal_3.html#T4" target="_self" title="XXREAL_3:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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