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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">H</font>, <font color="Olive" title="b2">N</font>, <font color="Olive" title="b3">N1</font>, <font color="Olive" title="b4">N2</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="b2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">N1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Olive" title="b4">N2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">N1</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Olive" title="b1">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">N2</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Olive" title="b1">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">N</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Olive" title="b1">H</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</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"> for <font color="Olive" title="b1">N</font>, <font color="Olive" title="b2">N1</font>, <font color="Olive" title="b3">N2</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">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Olive" title="b3">N2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">N1</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">N2</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</font>, <font color="Maroon" title="c4">N1</font>, <font color="Maroon" title="c5">N2</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="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c5">N2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">N2</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:63"/>
<font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c5">N2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">N2</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span><br/>

<a NAME="E2:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A1</font></span>: 
( <font color="Maroon" title="c3">N</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">N1</font> &amp; <font color="Maroon" title="c3">N</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c5">N2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T106" title="GROUP_2:th.106">GROUP_2:106</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">N2</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> or <font color="Maroon" title="c6">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#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:63"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">N2</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</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="c6">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#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font></span><br/>

<a NAME="E4:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A2</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N1</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N2</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:63"/>
( <font color="Maroon" title="c4">N1</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c5">N2</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_11.html#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="group_11.html#T60" target="_self" title="GROUP_11:th.60">Th60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:63"/>
<font color="Maroon" title="c6">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#K3" title="GROUP_11:func.3">`</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E37">A2</font></span></a></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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