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<a NAME="E1:1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st (  for <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] ) holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] ) implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">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">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E6">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_1"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">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">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]</span><br/><a NAME="E3:1_1_1_1"/>
<font color="Maroon" title="c1">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="ordinal1.html#T29" title="ORDINAL1:th.29">ORDINAL1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="#E1:1_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E1:1_1_1_1"><span class="lab"><font color="Green" title="E7">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><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c2">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><a NAME="E2:1_1_1_2"/><span class="lab"><font color="Green" title="E7">A9</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">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">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]</span><br/><a NAME="E3:1_1_1_2"/>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_2"><span class="lab"><font color="Green" title="E7">A9</font></span></a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E1:1_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E2:1_1_1_2"><span class="lab"><font color="Green" title="E7">A9</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">hence </span><a NAME="E4:1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E2:1_1_1"><span class="lab"><font color="Green" title="E6">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">thus </span><a NAME="E2:1_1"/>
 for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E4">A4</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/>


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