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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">l</font> be   <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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" title="b1">U</font> being   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st <font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Olive" title="b1">U</font> ) holds <br/> <a href="ordinal6.html#K8" title="ORDINAL6:func.8">lims</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> is   <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>   <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Olive" title="b1">U</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">U</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</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">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> ) implies  <a href="ordinal6.html#K8" title="ORDINAL6:func.8">lims</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> is   <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>   <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">G</font> = <font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a> ;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:119"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
<font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <span class="kw">and </span><br/><a NAME="E2:119"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ordinal6.html#K8" title="ORDINAL6:func.8">lims</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> is   <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>   <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font></span><br/>

<a NAME="E3:119"/>
 <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="c1">l</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="E4:119"/><span class="kw">then </span>
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D11" title="ORDINAL1:def.11">ORDINAL1:def 11</a></span>;<br/>
<a NAME="E5:119"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A3</font></span>: 
(  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> &amp; <font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<a NAME="E6:119"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A4</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</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="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></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="c2">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D15" target="_self" title="ORDINAL6:def.15">V</a></span>;<br/>
<a NAME="E7:119"/>
<font color="Maroon" title="c1">l</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:119"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E8:119"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A5</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="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:119"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</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="c5">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> is   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> )</span><br/><span class="kw">assume </span><a NAME="E1:119_1"/><span class="lab"><font color="Green" title="E67">A6</font></span>: 
<font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> is   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font></span><br/><a NAME="E2:119_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</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="E3:119_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> is   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="txt" href="#E1:119_1"><span class="lab"><font color="Green" title="E67">A6</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><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> =  <a href="ordinal6.html#K8" title="ORDINAL6:func.8">lims</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> as   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="ordinal6.html#T56" target="_self" title="ORDINAL6:th.56">ThL1</a></span>;<br/>
<a NAME="E11:119"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a>, <a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">g0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> as   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<a NAME="E13:119"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <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="c2">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/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> : <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> </span> } </span> ;<br/>
<a NAME="E14:119"/><span class="lab"><font color="Green" title="E68">A7</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</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="ordinal5.html#D2" title="ORDINAL5:def.2">ORDINAL5:def 2</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="c7">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> <font color="Maroon" title="c5">f</font> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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"> (  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font> or  not <font color="Maroon" title="c8">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> <font color="Maroon" title="c5">f</font> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:119_2"/><span class="lab"><font color="Green" title="E69">B1</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">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> <font color="Maroon" title="c5">f</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">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span><br/>

<a NAME="E2:119_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">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> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E3:119_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">B3</font></span>: 
( <font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">a</font>) &amp; <font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">b</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_2"><span class="lab"><font color="Green" title="E69">B1</font></span></a>, <a class="ref" href="ordinal6.html#D12" target="_self" title="ORDINAL6:def.12">LIM</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">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="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> or <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:119_2"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</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="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:119_2"/><span class="lab"><font color="Green" title="E71">B4</font></span>: 
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:119_2"><span class="lab"><font color="Green" title="E70">B3</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">xa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:119_2"/><span class="lab"><font color="Green" title="E72">B5</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xa</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c11">xa</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:119_2"><span class="lab"><font color="Green" title="E71">B4</font></span></a></span>;<br/>
<a NAME="E9:119_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xa</font> as   <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</font></span></a></span>;<br/>
<a NAME="E11:119_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font> <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="c2">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="E12:119_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:119"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E1:119_2"><span class="lab"><font color="Green" title="E69">B1</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/>
<a NAME="E13:119_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">b</font>
 
<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="E14:119_2"/><span class="kw">then </span>
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">b</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:119_2"><span class="lab"><font color="Green" title="E71">B4</font></span></a>, <a class="txt" href="#E8:119_2"><span class="lab"><font color="Green" title="E72">B5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:119_2"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:119_2"><span class="lab"><font color="Green" title="E70">B3</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E15:119"/>
<font color="Maroon" title="c5">f</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</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="c7">a</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> <font color="Maroon" title="c5">f</font> or <font color="Maroon" title="c7">a</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="c7">a</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> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> or <font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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"> (  not <font color="Maroon" title="c7">a</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> <font color="Maroon" title="c5">f</font> or <font color="Maroon" title="c7">a</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="c7">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  not <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> or <font color="Maroon" title="c8">b</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:119_3"/><span class="lab"><font color="Green" title="E69">C1</font></span>: 
( <font color="Maroon" title="c7">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> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c7">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="c7">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</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="c8">b</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font></span><br/>

<a NAME="E2:119_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D12" target="_self" title="ORDINAL6:def.12">LIM</a></span>;<br/>
<a NAME="E3:119_3"/><span class="lab"><font color="Green" title="E71">D1</font></span>: 
<font color="Maroon" title="c7">a</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> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_3"><span class="lab"><font color="Green" title="E69">C1</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E4:119_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">C3</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</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="E5:119_3"/><span class="lab"><font color="Green" title="E73">C4</font></span>: 
<font color="Maroon" title="c8">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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:119_3_1"/>
<font color="Maroon" title="c8">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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">b</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font> or <font color="Maroon" title="c9">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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:119_3_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">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="c9">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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:119_3_1_1"/><span class="lab"><font color="Green" title="E74">B4</font></span>: 
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119_3"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">xa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:119_3_1_1"/><span class="lab"><font color="Green" title="E75">B5</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xa</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c11">xa</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:119_3_1_1"><span class="lab"><font color="Green" title="E74">B4</font></span></a></span>;<br/>
<a NAME="E6:119_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xa</font> as   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</font></span></a></span>;<br/>
<a NAME="E8:119_3_1_1"/><span class="lab"><font color="Green" title="E76">C6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font> <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="c2">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="E9:119_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">C5</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:119"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E1:119_3"><span class="lab"><font color="Green" title="E69">C1</font></span></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></span>;<br/>
<a NAME="E10:119_3_1_1"/><span class="lab"><font color="Green" title="E78">C7</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a></span>;<br/>
<a NAME="E11:119_3_1_1"/><span class="lab"><font color="Green" title="E79">C8</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="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:119"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E3:119_3"><span class="lab"><font color="Green" title="E71">D1</font></span></a>, <a class="txt" href="#E8:119_3_1_1"><span class="lab"><font color="Green" title="E76">C6</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E12:119_3_1_1"/><span class="kw">then </span>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_3"><span class="lab"><font color="Green" title="E69">C1</font></span></a>, <a class="txt" href="#E10:119_3_1_1"><span class="lab"><font color="Green" title="E78">C7</font></span></a>, <a class="ref" href="ordinal5.html#T6" title="ORDINAL5:th.6">ORDINAL5:6</a></span>;<br/>
<a NAME="E13:119_3_1_1"/><span class="kw">then </span>
 <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <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"><font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#D10" title="ORDINAL2:def.10">ORDINAL2:def 10</a></span>;<br/>
<a NAME="E14:119_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</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"><font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:119_3_1_1"><span class="lab"><font color="Green" title="E77">C5</font></span></a>, <a class="ref" href="ordinal2.html#D10" title="ORDINAL2:def.10">ORDINAL2:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:119_3_1_1"/><span class="lab"><font color="Green" title="E80">C9</font></span>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:119_3_1_1"><span class="lab"><font color="Green" title="E74">B4</font></span></a>, <a class="txt" href="#E5:119_3_1_1"><span class="lab"><font color="Green" title="E75">B5</font></span></a>, <a class="txt" href="#E11:119_3_1_1"><span class="lab"><font color="Green" title="E79">C8</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>;<br/>
<a NAME="E17:119_3_1_1"/><span class="lab"><font color="Green" title="E81">D3</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:119"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E1:119_3"><span class="lab"><font color="Green" title="E69">C1</font></span></a>, <a class="txt" href="#E16:119_3_1_1"><span class="lab"><font color="Green" title="E80">C9</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E18:119_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c13">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:119_3_1_1"><span class="lab"><font color="Green" title="E80">C9</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/>
<a NAME="E19:119_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:119_3_1_1"><span class="lab"><font color="Green" title="E75">B5</font></span></a></span>;<br/>
<a NAME="E20:119_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:119_3_1_1"><span class="lab"><font color="Green" title="E80">C9</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E21:119_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c13">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:119"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E17:119_3_1_1"><span class="lab"><font color="Green" title="E81">D3</font></span></a>, <a class="ref" href="ordinal6.html#D12" target="_self" title="ORDINAL6:def.12">LIM</a></span>;<br/>
<a NAME="E22:119_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:119_3_1_1"><span class="lab"><font color="Green" title="E80">C9</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="E23:119_3_1_1"/>
<font color="Maroon" title="c9">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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_3"><span class="lab"><font color="Green" title="E72">C3</font></span></a>, <a class="txt" href="#E16:119_3_1_1"><span class="lab"><font color="Green" title="E80">C9</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="c9">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="c9">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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> or <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:119_3_1"/>
<font color="Maroon" title="c9">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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">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"> <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:119_3_1"/><span class="lab"><font color="Green" title="E74">E1</font></span>: 
( <font color="Maroon" title="c10">x</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>;<br/>
<a NAME="E5:119_3_1"/><span class="lab"><font color="Green" title="E75">E2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_3"><span class="lab"><font color="Green" title="E72">C3</font></span></a>, <a class="txt" href="#E4:119_3_1"><span class="lab"><font color="Green" title="E74">E1</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/>
<a NAME="E6:119_3_1"/>
<font color="Maroon" title="c10">x</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> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_3_1"><span class="lab"><font color="Green" title="E74">E1</font></span></a>, <a class="ref" href="relat_1.html#T57" title="RELAT_1:th.57">RELAT_1:57</a></span>;<br/>
<a NAME="E7:119_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D12" target="_self" title="ORDINAL6:def.12">LIM</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:119_3_1"/><span class="lab"><font color="Green" title="E76">E4</font></span>: 
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">x</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_3_1"><span class="lab"><font color="Green" title="E74">E1</font></span></a>, <a class="txt" href="#E5:119_3_1"><span class="lab"><font color="Green" title="E75">E2</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:119_3_1"/><span class="lab"><font color="Green" title="E77">E5</font></span>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c12">z</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:119_3_1"><span class="lab"><font color="Green" title="E76">E4</font></span></a></span>;<br/>
<a NAME="E12:119_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> as   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="#E8:119"><span class="lab"><font color="Green" title="E66">A5</font></span></a></span>;<br/>
<a NAME="E14:119_3_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">h</font> <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="c2">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="E15:119_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_3_1"><span class="lab"><font color="Green" title="E74">E1</font></span></a>, <a class="txt" href="#E4:119_3"><span class="lab"><font color="Green" title="E72">C3</font></span></a>, <a class="txt" href="#E1:119_3"><span class="lab"><font color="Green" title="E69">C1</font></span></a>, <a class="txt" href="#E13:119"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/>
<a NAME="E16:119_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c10">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font>
 
<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="E17:119_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c13">h</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">a</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:119_3_1"><span class="lab"><font color="Green" title="E76">E4</font></span></a>, <a class="txt" href="#E11:119_3_1"><span class="lab"><font color="Green" title="E77">E5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:119_3_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119_3"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E6:119_3"/>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font> is  <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:119"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="ref" href="ordinal6.html#T13" target="_self" title="ORDINAL6:th.13">ThND</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:119_3"/>
<font color="Maroon" title="c8">b</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_3"><span class="lab"><font color="Green" title="E69">C1</font></span></a>, <a class="txt" href="#E4:119_3"><span class="lab"><font color="Green" title="E72">C3</font></span></a>, <a class="txt" href="#E5:119_3"><span class="lab"><font color="Green" title="E73">C4</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:119"/>
 <a href="ordinal6.html#K8" title="ORDINAL6:func.8">lims</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> is   <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>   <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:119"><span class="lab"><font color="Green" title="E68">A7</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>
