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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b2">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Olive" title="b1">g</font></span><br/><b>let </b><font color="Maroon" title="c2">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c2">g</font> <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="c2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">g</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c2">g</font> <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="c2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">g</font></span><br/>





<a NAME="E1:48"/>
<font color="Maroon" title="c3">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c2">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="group_3.html#T40" target="_self" title="GROUP_3:th.40">Th40</a></i>;<br/>
<b>hence </b><a NAME="E2:48"/>
<font color="Maroon" title="c3">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c2">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="group_2.html#T6" title="GROUP_2:th.6">GROUP_2:6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c2">g</font></span><br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">g</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c2">g</font> )</span><br/>

<b>assume </b><a NAME="E3:48"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c2">g</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:48"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:48"/><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="#E5:48"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="group_2.html#T33" title="GROUP_2:th.33">GROUP_2:33</a></i>;<br/>
<a NAME="E8:48"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="#E5:48"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E7:48"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:48"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="#E7:48"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="group_3.html#T50" target="_self" title="GROUP_3:th.50">Th50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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