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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <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">phi</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st <font color="Olive" title="b1">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> holds <br/><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Olive" title="b1">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a></span><br/><b>let </b><font color="Maroon" title="c2">phi</font> be   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> implies <font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> )</span><br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a></span><br/>

<b>set </b><font color="Maroon" title="c3">xi</font> = <font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font>;<br/>
<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c5">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> or  not <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font> )</span><br/>



<b>assume </b><a NAME="E2:18"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font></span><br/>

<a NAME="E3:18"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">phi</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A'</font> = <font color="Maroon" title="c2">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c7">B'</font> = <font color="Maroon" title="c2">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E5:18"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c6">A'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">B'</font> &amp; <font color="Maroon" title="c6">A'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B'</font> )
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E3:18"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a>, <a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T49" title="ORDINAL2:th.49">ORDINAL2:49</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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