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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  implies for <font color="Olive" title="b1">fi</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">B</font> ) holds <br/> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Olive" title="b1">fi</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:34"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">fi</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">B</font> ) holds <br/> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Olive" title="b1">fi</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">fi</font> be   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">B</font> ) implies  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">fi</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:34"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E5:34"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">fi</font></span><br/>

<a NAME="E6:34"/>
<font color="Maroon" title="c3">fi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="ordinal4.html#T25" target="_self" title="ORDINAL4:th.25">Th25</a></span>;<br/>
<a NAME="E7:34"/><span class="kw">then </span>
 <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c3">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">fi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="ordinal4.html#T8" target="_self" title="ORDINAL4:th.8">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:34"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">fi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="ordinal2.html#T62" title="ORDINAL2:th.62">ORDINAL2:62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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