<?xml version="1.0"?>
<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="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:49"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="group_2.html#E1:49"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E5:49"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K12">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:49"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E7:49"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:49_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:49_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E6:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="group_2.html#E3:49_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E7:49_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E6:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="group_2.html#E3:49_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="group_2.html#E6:49_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="group_2.html#E6:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="group_2.html#E3:49_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="group_2.html#E6:49_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon">c<sub>5</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="group_2.html#E3:49_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="group_2.html#E4:49_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="group_2.html#E6:49_1"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>

;<br/>
<b>hence </b><a NAME="E10:49_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E2:49"><i><font color="Green">E35</font></i></a>, <a class="txt" href="group_2.html#E7:49_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="group_2.html#E6:49"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><i><font color="Green">E40</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>

<br/>.= 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>

;<br/><b>thus </b><a NAME="E3:49_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E1:49_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="group_2.html#E2:49_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:49"/>
 <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="group_1.html#V3">Group-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="group_2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E3:49_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="group_2.html#E3:49"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E4:49_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="group_2.html#E2:49"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) <b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="group_2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_2.html#E9:49"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="group_2.html#E9:49"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K3">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) <b>by </b><i><a class="txt" href="group_2.html#E3:49"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_2.html#E9:49"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="group_2.html#E9:49"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="group_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E13:49"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
