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<b>let </b><font color="Maroon" title="c1">ADG</font> be   <a href="afvect0.html#NM2" title="AFVECT0:NM.2">Proper_Uniquely_Two_Divisible_Group</a>;<br/><b>let </b><font color="Maroon" title="c2">o'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font>;<br/><b>let </b><font color="Maroon" title="c3">o</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o'</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">AS</font> =  <a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font>;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Y</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">Z</font> =  <a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font>;<br/>
<b>set </b><font color="Maroon" title="c8">T</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font> = <font color="Maroon" title="c2">o'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c9">g</font> being   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font><b> such that </b><br/><a NAME="E3:79"/><i><font color="Green" title="E51">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font> holds  <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E4:79"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font> <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 <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font> <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 <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></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 <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tdgroup.html#T9" title="TDGROUP:th.9">TDGROUP:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font></span>)</span> ;<br/>
<a NAME="E6:79"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="afvect0.html#E3:79"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="afvect0.html#T76" target="_self" title="AFVECT0:th.76">Th76</a></i>;<br/>
<a NAME="E7:79"/><i><font color="Green" title="E53">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">f</font> <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 <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="afvect0.html#E3:79"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="afvect0.html#T77" target="_self" title="AFVECT0:th.77">Th77</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font>;<br/><b>thus </b><a NAME="E1:79_1"/>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="afvect0.html#E1:79"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="afvect0.html#E3:79"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="afvect0.html#T75" target="_self" title="AFVECT0:th.75">Th75</a></i>;<br/><b>thus </b><a NAME="E2:79_1"/>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">ADG</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="afvect0.html#E1:79"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="afvect0.html#E3:79"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="afvect0.html#T75" target="_self" title="AFVECT0:th.75">Th75</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">fa</font> = <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span> ;<br/><b>thus </b><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="afvect0.html#K4" title="AFVECT0:func.4">Pcom</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">fa</font>
<b>by </b><i><a class="txt" href="afvect0.html#E1:79"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="afvect0.html#E3:79"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="afvect0.html#T75" target="_self" title="AFVECT0:th.75">Th75</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="afvect0.html#T58" target="_self" title="AFVECT0:th.58">Th58</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E9:79"/><b>then </b>
<font color="Maroon" title="c10">f</font> <a href="afvect0.html#R3" title="AFVECT0:pred.3">is_Iso_of</a> <font color="Maroon" title="c1">ADG</font>, <a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font>
 
<b>by </b><i><a class="txt" href="afvect0.html#E6:79"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="afvect0.html#E7:79"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="afvect0.html#D10" target="_self" title="AFVECT0:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E10:79"/>
<font color="Maroon" title="c1">ADG</font>, <a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>,<font color="Maroon" title="c3">o</font> <a href="afvect0.html#R4" title="AFVECT0:pred.4">are_Iso</a> 
 <b>by </b><i><a class="ref" href="afvect0.html#D11" target="_self" title="AFVECT0:def.11">Def11</a></i>;<br/>


</div>
