<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="sheffer1.html#L1" title="SHEFFER1:struct.1">ShefferStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> implies ( <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V10" title="SHEFFER1:attr.10">satisfying_Sheffer_1</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V11" title="SHEFFER1:attr.11">satisfying_Sheffer_2</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V12" title="SHEFFER1:attr.12">satisfying_Sheffer_3</a> ) )</span><br/>

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V10" title="SHEFFER1:attr.10">satisfying_Sheffer_1</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V11" title="SHEFFER1:attr.11">satisfying_Sheffer_2</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V12" title="SHEFFER1:attr.12">satisfying_Sheffer_3</a> )</span><br/>

<b>thus </b><a NAME="E2:45_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V10" title="SHEFFER1:attr.10">satisfying_Sheffer_1</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V11" title="SHEFFER1:attr.11">satisfying_Sheffer_2</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V12" title="SHEFFER1:attr.12">satisfying_Sheffer_3</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">SHEFFER1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font></span><br/>

<b>thus </b><a NAME="E1:45_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:45_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V11" title="SHEFFER1:attr.11">satisfying_Sheffer_2</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V12" title="SHEFFER1:attr.12">satisfying_Sheffer_3</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">SHEFFER1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span><br/>

<b>thus </b><a NAME="E1:45_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="sheffer1.html#D15" target="_self" title="SHEFFER1:def.15">SHEFFER1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span></span><br/>

<b>thus </b><a NAME="E4:45_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
