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<div class="add">

<b>consider </b><font color="Maroon" title="c2">H</font> being  <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/>
<b>set </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">H</font>;<br/>
<a NAME="E2:17_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E3:17_1_1"/>
<font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">H</font> is <a href="group_7.html#V1" title="GROUP_7:attr.1">HGrStr-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</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="group_7.html#D1" target="_self" title="GROUP_7:def.1">GROUP_7:def 1</a><br/>

<b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon" title="c4">i</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"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">H</font></span>)</span>
 ;<br/>

<a NAME="E2:17_1_1_1"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</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"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">H</font></span>)</span> &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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/>
<a NAME="E3:17_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:17_1_1_1"/>
<font color="Maroon" title="c4">i</font> is  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> 
 <b>by </b><i><a class="txt" href="group_7.html#E2:17_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="group_7.html#E2:17_1_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">H</font> as   <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">f</font>
;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="group_7.html#D3" target="_self" title="GROUP_7:def.3">GROUP_7:def 3</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:17_1_1_2"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">I1</font> = <font color="Maroon" title="c1">I</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c7">i1</font> = <font color="Maroon" title="c5">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">I1</font> <b>by </b><i><a class="txt" href="group_7.html#E1:17_1_1_2"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">F1</font> = <font color="Maroon" title="c4">f</font> as   <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <font color="Maroon" title="c6">I1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">Fi</font> = <font color="Maroon" title="c8">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c7">i1</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  <b>by </b><i><a class="txt" href="group_7.html#E2:17_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c10">Fi</font> = <font color="Maroon" title="c9">Fi</font>;<br/><b>thus </b><a NAME="E6:17_1_1_2"/>
<font color="Maroon" title="c10">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="group_7.html#D4" target="_self" title="GROUP_7:def.4">GROUP_7:def 4</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:17_1_1_3"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">I1</font> = <font color="Maroon" title="c1">I</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c7">i1</font> = <font color="Maroon" title="c5">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">I1</font> <b>by </b><i><a class="txt" href="group_7.html#E1:17_1_1_3"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">F1</font> = <font color="Maroon" title="c4">f</font> as   <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <font color="Maroon" title="c6">I1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">Fi</font> = <font color="Maroon" title="c8">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c7">i1</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  <b>by </b><i><a class="txt" href="group_7.html#E2:17_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c10">Fi</font> = <font color="Maroon" title="c9">Fi</font>;<br/><b>thus </b><a NAME="E6:17_1_1_3"/>
<font color="Maroon" title="c10">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">i</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="group_7.html#D5" target="_self" title="GROUP_7:def.5">GROUP_7:def 5</a><br/>

<b>assume </b><a NAME="E7:17_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">I1</font> = <font color="Maroon" title="c1">I</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">i1</font> = <font color="Maroon" title="c5">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">I1</font> <b>by </b><i><a class="txt" href="group_7.html#E7:17_1_1"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">F1</font> = <font color="Maroon" title="c4">f</font> as   <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <font color="Maroon" title="c6">I1</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Fi</font> = <font color="Maroon" title="c8">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c7">i1</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  <b>by </b><i><a class="txt" href="group_7.html#E2:17_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">Fi</font>
;<br/>

<b>thus </b><a NAME="E12:17_1_1"/>
<font color="Maroon" title="c9">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ;<br/>


</div>
