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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="sheffer1.html#V10">satisfying_Sheffer_1</a> <a href="sheffer1.html#V11">satisfying_Sheffer_2</a> <a href="sheffer1.html#V12">satisfying_Sheffer_3</a>  <a href="sheffer1.html#L1">ShefferStr</a> ;<br/>

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:166"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5">|</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="sheffer1.html#K5">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sheffer2.html#D1" target="_self">SHEFFER2:def 1</a><br/>

<a NAME="E2:166"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">a</font> <a href="sheffer1.html#K5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="sheffer1.html#K5">|</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="sheffer2.html#T107" target="_self">Th107</a></i>;<br/>
<a NAME="E3:166"/>
not <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5">|</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5">|</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="sheffer2.html#T143" target="_self">Th143</a>, <a class="ref" href="sheffer2.html#T68" target="_self">Th68</a></i>;<br/>
<b>hence </b><a NAME="E4:166"/>
contradiction
 <b>by </b><i><a class="ref" href="sheffer2.html#T144" target="_self">Th144</a>, <a class="ref" href="sheffer2.html#T141" target="_self">Th141</a></i>;<br/>


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