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<b>let </b><font color="Maroon" title="c1">H</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">H</font>;<br/><b>let </b><font color="Maroon" title="c3">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font>;<br/>





<b>set </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">h</font> <a href="topgrp_1.html#K1" title="TOPGRP_1:func.1">*</a> ;<br/>
<a NAME="E1:20"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="topgrp_1.html#K1" title="TOPGRP_1:func.1">*</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="topgrp_1.html#K1" title="TOPGRP_1:func.1">*</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="topgrp_1.html#K1" title="TOPGRP_1:func.1">*</a> </span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="topgrp_1.html#K1" title="TOPGRP_1:func.1">*</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font> <b>by </b><i><a class="txt" href="topgrp_1.html#E3:20_1"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/><a NAME="E5:20_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="topgrp_1.html#K1" title="TOPGRP_1:func.1">*</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#D1" target="_self" title="TOPGRP_1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E6:20_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">h</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E3:20_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="group_2.html#T33" title="GROUP_2:th.33">GROUP_2:33</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">y</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="E3:20"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">h</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c2">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font><b> such that </b><br/><a NAME="E5:20"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T33" title="GROUP_2:th.33">GROUP_2:33</a></i>;<br/>
<a NAME="E6:20"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="topgrp_1.html#K1" title="TOPGRP_1:func.1">*</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#D1" target="_self" title="TOPGRP_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E7:20"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="topgrp_1.html#K1" title="TOPGRP_1:func.1">*</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="topgrp_1.html#E5:20"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


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