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<b>let </b><font color="Maroon" title="c4">x</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> ;<br/>

<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ;<br/>

<a NAME="E2:69"/><b>then </b><i><font color="Green" title="E62">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E1:69"><i><font color="Green" title="E61">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E1:69"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="arytm_2.html#E11"><i><font color="Green" title="E7">Lm6</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">y</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> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:69_1"/><i><font color="Green" title="E63">A3</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font>
 ;<br/><b>hence </b><a NAME="E2:69_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:69"><i><font color="Green" title="E62">A2</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E11"><i><font color="Green" title="E7">Lm6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E2:69"><i><font color="Green" title="E62">A2</font></i></a>, <a class="txt" href="arytm_2.html#E1:69_1"><i><font color="Green" title="E63">A3</font></i></a></i>;<br/><b>thus </b><a NAME="E5:69_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:69_1"><i><font color="Green" title="E63">A3</font></i></a></i>;<br/><a NAME="E6:69_1"/>
<font color="Maroon" title="c9">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:69_1"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:69_1"/><i><font color="Green" title="E64">A4</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T30" title="ORDINAL3:th.30">ORDINAL3:30</a></i>;<br/><a NAME="E9:69_1"/>
<font color="Maroon" title="c10">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:69_1"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c10">C</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/><a NAME="E11:69_1"/>
<font color="Maroon" title="c11">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">z'</font> = <font color="Maroon" title="c11">C</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E11"><i><font color="Green" title="E7">Lm6</font></i></a></i>;<br/><a NAME="E13:69_1"/>
<font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">z'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:69_1"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="arytm_2.html#E75"><i><font color="Green" title="E58">Lm46</font></i></a></i>;<br/><a NAME="E14:69_1"/><b>then </b>
<font color="Maroon" title="c8">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">x'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/><b>hence </b><a NAME="E15:69_1"/>
<font color="Maroon" title="c7">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green" title="E19">Lm15</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:69"/><i><font color="Green" title="E63">A5</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E6:69"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="arytm_2.html#E11"><i><font color="Green" title="E7">Lm6</font></i></a></i>;<br/>
<b>assume </b><a NAME="E9:69"/><i><font color="Green" title="E64">A6</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 ;<br/>

<b>assume </b><a NAME="E10:69"/>
<font color="Maroon" title="c7">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">x</font>
 ;<br/>

<a NAME="E11:69"/><b>then </b>
<font color="Maroon" title="c9">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">x'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green" title="E19">Lm15</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">z'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E13:69"/><i><font color="Green" title="E65">A7</font></i>: 
<font color="Maroon" title="c9">y'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c10">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c10">z'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E1:69"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="arytm_2.html#E6:69"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="arytm_2.html#E13:69"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="arytm_3.html#T78" title="ARYTM_3:th.78">ARYTM_3:78</a></i>;<br/>
<a NAME="E15:69"/>
<font color="Maroon" title="c8">n</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">+^</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E13:69"><i><font color="Green" title="E65">A7</font></i></a>, <a class="txt" href="arytm_2.html#E75"><i><font color="Green" title="E58">Lm46</font></i></a></i>;<br/>
<a NAME="E16:69"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></i>;<br/>
<a NAME="E17:69"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E9:69"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E18:69"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>


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