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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">N</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Olive" title="b1">A</font></span>)</span> holds <br/><font color="Olive" title="b2">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span> holds <br/><font color="Olive" title="b1">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span> implies <font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</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">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:38"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E4:38"/>
<font color="Maroon" title="c5">x</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="group_11.html#T14" target="_self" title="GROUP_11:th.14">Th14</a></span>;<br/>
<a NAME="E5:38"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K1" title="GROUP_11:func.1">`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="group_11.html#K2" title="GROUP_11:func.2">~</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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