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<b>let </b><font color="Maroon">MS</font> be   <a href="analmetr.html#NM2">OrtAfPl</a>;<br/>

<b>set </b><font color="Maroon">AS</font> =  <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">MS</font> is <a href="euclmetr.html#V2" target="_self">Pappian</a>
 ;<br/><a NAME="E2:18_1"/><b>then </b>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font> <a href="papdesaf.html#NR1" target="_self">satisfies_PAP'</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_1"/>
 <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">MS</font>
 <b>by </b><i><a class="ref" href="euclmetr.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2"/>
 <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">MS</font>
 ;<br/><a NAME="E2:18_2"/><b>then </b>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font> is <a href="aff_2.html#V2">Pappian</a>
 <b>by </b><i><a class="ref" href="euclmetr.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:18_2"/>
<font color="Maroon">MS</font> is <a href="euclmetr.html#V2" target="_self">Pappian</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
( <font color="Maroon">MS</font> is <a href="euclmetr.html#V2" target="_self">Pappian</a> iff  <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">MS</font> )
 <b>by </b><i/>;<br/>


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