<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c6">z</font> implies <font color="Maroon" title="c6">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">x</font> )</span><br/>





<a NAME="E1:70"/>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">zz</font> =  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#T1" target="_self" title="ARYTM_2:th.1">Th1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:70"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c6">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">x</font></span><br/>

<span class="kw">assume </span><a NAME="E4:70"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
 not <font color="Maroon" title="c6">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <span class="kw"> such that </span><br/><a NAME="E6:70"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#T9" target="_self" title="ARYTM_2:th.9">Th10</a></span>;<br/>
<a NAME="E7:70"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">y0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="arytm_2.html#T6" target="_self" title="ARYTM_2:th.6">Th7</a></span>;<br/>
<a NAME="E8:70"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">y0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></span>;<br/>
<a NAME="E9:70"/><span class="kw">then </span>
<font color="Maroon" title="c8">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#T5" target="_self" title="ARYTM_2:th.5">Th6</a>, <a class="ref" href="arytm_2.html#T11" target="_self" title="ARYTM_2:th.11">Th12</a></span>;<br/>
<a NAME="E10:70"/><span class="kw">then </span>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:70"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E61">A2</font></span></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>
