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



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

<span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</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> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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> <font color="Maroon" title="c4">B</font> or  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">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">xi</font> = <font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E3:18"/><span class="lab"><font color="Green" title="E16">A3</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">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">A9</font> = <font color="Maroon" title="c2">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c7">B9</font> = <font color="Maroon" title="c2">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E5:18"/><span class="lab"><font color="Green" title="E17">A4</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">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D1" title="ORDINAL3:def.1">ORDINAL3:def 1</a></span>;<br/>
<a NAME="E6:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="ordinal3.html#D1" title="ORDINAL3:def.1">ORDINAL3:def 1</a></span>;<br/>
<a NAME="E7:18"/>
<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"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E8:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c6">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="ordinal3.html#D1" title="ORDINAL3:def.1">ORDINAL3:def 1</a></span>;<br/>
<a NAME="E9:18"/>
<font color="Maroon" title="c6">A9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c2">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="ordinal2.html#T32" title="ORDINAL2:th.32">ORDINAL2:32</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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