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<span class="kw">set </span><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E1:81_1_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:81_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></span>;<br/>
<a NAME="E3:81_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">B2</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<a NAME="E4:81_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
;<br/>
<a NAME="E5:81_1_1"/><span class="kw">then </span>
1 <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E6:81_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T11" title="ORDINAL1:th.11">ORDINAL1:11</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E41">S4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</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"> ( <font color="Maroon" title="c4">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">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c4">b</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_1"/>
<font color="Maroon" title="c4">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">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c4">b</font>)</span><br/><a NAME="E2:81_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:81_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c4">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D8" target="_self" title="ORDINAL6:def.8">EXP</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>
<span class="kw">hence </span><a NAME="E8:81_1_1"/>
<font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</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="#E6:81_1_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="ordinal5.html#T10" title="ORDINAL5:th.10">ORDINAL5:10</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal6.html#D3" target="_self" title="ORDINAL6:def.3">ORDINAL6:def 3</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="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">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">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c4">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> or <font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">c</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</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"> (  not <font color="Maroon" title="c4">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">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c4">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  not <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c5">c</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E9:81_1_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
( <font color="Maroon" title="c4">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">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Maroon" title="c4">b</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="c4">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</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="c5">c</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">b</font></span><br/>

<a NAME="E10:81_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E11:81_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E12:81_1_1"/><span class="lab"><font color="Green" title="E44">S7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</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="#E7:81_1_1"><span class="lab"><font color="Green" title="E41">S4</font></span></a>, <a class="txt" href="#E6:81_1_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a>, <a class="ref" href="ordinal5.html#T10" title="ORDINAL5:th.10">ORDINAL5:10</a></span>;<br/>
<a NAME="E13:81_1_1"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E9:81_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<a NAME="E14:81_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c4">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D8" target="_self" title="ORDINAL6:def.8">EXP</a></span>;<br/>
<a NAME="E15:81_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:81_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">c</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:81_1_1_2_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">d</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:81_1_1_2_1"/><span class="lab"><font color="Green" title="E47">B1</font></span>: 
( <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">d</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1"><span class="lab"><font color="Green" title="E39">B2</font></span></a>, <a class="txt" href="#E9:81_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E14:81_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="ordinal5.html#T11" title="ORDINAL5:th.11">ORDINAL5:11</a></span>;<br/>
<a NAME="E4:81_1_1_2_1"/>
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:81_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E3:81_1_1_2_1"><span class="lab"><font color="Green" title="E47">B1</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">then </span> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">d</font>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D8" target="_self" title="ORDINAL6:def.8">EXP</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_2_1"><span class="lab"><font color="Green" title="E47">B1</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:81_1_1_2_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81_1_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E3:81_1_1_2_1"><span class="lab"><font color="Green" title="E47">B1</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</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>

<span class="kw">let </span><font color="Maroon" title="c6">c</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span> or <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:81_1_1_2"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span>
 ; <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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:81_1_1_2"/><span class="lab"><font color="Green" title="E47">B3</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81_1_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81_1_1_2"><span class="lab"><font color="Green" title="E47">B3</font></span></a></span>;<br/>
<a NAME="E6:81_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:81_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E4:81_1_1_2"><span class="lab"><font color="Green" title="E47">B3</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">then </span> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">x</font>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D8" target="_self" title="ORDINAL6:def.8">EXP</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81_1_1_2"><span class="lab"><font color="Green" title="E47">B3</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:81_1_1_2"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1"><span class="lab"><font color="Green" title="E39">B2</font></span></a>, <a class="txt" href="#E4:81_1_1_2"><span class="lab"><font color="Green" title="E47">B3</font></span></a>, <a class="txt" href="#E9:81_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E14:81_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="ordinal5.html#T11" title="ORDINAL5:th.11">ORDINAL5:11</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>
<span class="kw">hence </span><a NAME="E16:81_1_1"/>
<font color="Maroon" title="c5">c</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">U</font> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:81_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E12:81_1_1"><span class="lab"><font color="Green" title="E44">S7</font></span></a>, <a class="txt" href="#E11:81_1_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="ordinal5.html#T6" title="ORDINAL5:th.6">ORDINAL5:6</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>
