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<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>, <font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>;<br/>





<a NAME="E1:2"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<b>let </b><font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E2:2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 ;<br/>

<a NAME="E3:2"/>
 <a href="relat_1.html#K1">dom</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:2"/><b>then </b><i><font color="Green">E41</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:2"/><i><font color="Green">E42</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E6:2"/>
<font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b2</font>
 
<b>by </b><i>, , , , <a class="ref" href="altcat_1.html#T7">ALTCAT_1:7</a></i>;<br/>
<b>hence </b><a NAME="E7:2"/>
<font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b2</font>
 ;<br/>


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