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<div class="add">

<b>set </b><font color="Maroon" title="c1">op</font> = the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c2">carr</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E1:95_1"/><i><font color="Green" title="E48">A2</font></i>: 
( <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:95_1"/><b>then </b><i><font color="Green" title="E49">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E3:95_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:95_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <b>by </b><i><a class="txt" href="monoid_0.html#E1:95_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="monoid_0.html#E3:95_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c5">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> , <font color="Maroon" title="c7">y2</font> = <font color="Maroon" title="c5">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<a NAME="E6:95_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E7:95_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y2</font>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E3:95_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E8:95_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:95"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">f</font> = the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">F<sub>2</sub></font>() as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="monoid_0.html#E2:95_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E5:95_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">H</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">f</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">Def23</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">H</font>
;<br/>

<b>thus </b><a NAME="E7:95_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>


</div>
