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<span class="kw">let </span><font color="Maroon" title="c1">g</font> be   <a href="ordinal6.html#V3" title="ORDINAL6:attr.3">Ordinal-Sequence-valued</a>  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">g</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; (  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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">g</font> holds <br/><font color="Maroon" title="c1">g</font> <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>  ) implies  <a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:100"/><span class="lab"><font color="Green" title="E45">Z0</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">g</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <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 <br/>( <font color="Olive" title="b1">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="c1">g</font> &amp;  not <font color="Maroon" title="c1">g</font> <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>  ) or  <a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:100"/><span class="lab"><font color="Green" title="E46">Z1</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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">g</font> holds <br/><font color="Maroon" title="c1">g</font> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">f</font> =  <a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">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="c3">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> or <font color="Maroon" title="c3">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="c3">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</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"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">a</font> )</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> ; <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="c3">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> or <font color="Maroon" title="c3">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="c3">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> or <font color="Maroon" title="c4">b</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">a</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">h</font> = <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">a</font> as   <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</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/>
<span class="kw">assume </span><a NAME="E4:100"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
( <font color="Maroon" title="c3">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"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Maroon" title="c3">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="c3">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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="c4">b</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">a</font></span><br/>

<a NAME="E5:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<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> <font color="Maroon" title="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E45">Z0</font></span></a>, <a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E46">Z1</font></span></a>, <a class="ref" href="ordinal6.html#T52" target="_self" title="ORDINAL6:th.52">ThA9</a></span>;<br/>
<a NAME="E6:100"/>
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E7:100"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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/>
<span class="kw">hence </span><a NAME="E8:100"/>
<font color="Maroon" title="c4">b</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E5:100"><span class="lab"><font color="Green" title="E48">A2</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/>


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