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<span class="kw">let </span><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st <font color="Olive" title="b1">G1</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">G2</font> &amp; <font color="Olive" title="b2">G2</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">G3</font> holds <br/><font color="Olive" title="b1">G1</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">G3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G1</font>, <font color="Maroon" title="c3">G2</font>, <font color="Maroon" title="c4">G3</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">G1</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font> &amp; <font color="Maroon" title="c3">G2</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">G3</font> implies <font color="Maroon" title="c2">G1</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">G3</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c2">G1</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font>
 <span class="kw">and </span><br/><a NAME="E2:72"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c3">G2</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">G3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">G1</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">G3</font></span><br/>

<a NAME="E3:72"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c2">G1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E54">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">G1</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G3</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G1</font></span><br/><a NAME="E1:72_1"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/><font color="Maroon" title="c2">G1</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c5">o</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G2</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G3</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G3</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:72_1"/>
<font color="Maroon" title="c2">G1</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G3</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:72"/>
<font color="Maroon" title="c3">G2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/>
<a NAME="E6:72"/><span class="kw">then </span>
<font color="Maroon" title="c2">G1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="group_2.html#T56" title="GROUP_2:th.56">GROUP_2:56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:72"/>
<font color="Maroon" title="c2">G1</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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