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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="monoid_0.html#M3" target="_self">MonoidalSubStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:74"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D23" target="_self">Def23</a>, <a class="ref" href="monoid_0.html#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E2:74"/><b>then </b><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<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="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<b>thus </b><a NAME="E3:74"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:74_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 ;<br/>

<a NAME="E2:74_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 
<b>by </b><i><a class="txt" href="monoid_0.html#E2:74"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:74_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:74"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</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="E4:74"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 ;<br/>

<a NAME="E5:74"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 
<b>by </b><i><a class="txt" href="monoid_0.html#E2:74"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E6:74"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:74"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


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