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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:72"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:72_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E2:72_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_4.html#T40" target="_self">Th40</a></i>
;<br/>


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

<b>thus </b><a NAME="E2:72"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="group_4.html#T78" target="_self">Th78</a></i>;<br/>


</div>
