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<b>let </b><font color="Maroon" title="c1">D</font>, <font color="Maroon" title="c2">C</font> be  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>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">f'</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c5">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon" title="c5">F</font> <a href="finseqop.html#NR2" title="FINSEQOP:NR.2">has_an_inverseOp</a>  &amp; <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  )
 <b>and </b><br/><a NAME="E2:86"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">e</font> =  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c5">F</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c2">C</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="funcop_1.html#T58" title="FUNCOP_1:th.58">FUNCOP_1:58</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">u</font> =  <a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c5">F</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>;<br/><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font></span>)</span> = 
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:86"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="funcop_1.html#T48" title="FUNCOP_1:th.48">FUNCOP_1:48</a></i>
<br/>.= 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c5">F</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font> = 
<span class="p1">(<span class="default"><a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:86"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="finseqop.html#T64" target="_self" title="FINSEQOP:th.64">Th64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:86"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>;<br/><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font></span>)</span> = 
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:86"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="funcop_1.html#T48" title="FUNCOP_1:th.48">FUNCOP_1:48</a></i>
<br/>.= 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c5">F</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="E2:86_2"/><b>then </b>
<font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseqop.html#E1:86"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="finseqop.html#T64" target="_self" title="FINSEQOP:th.64">Th64</a></i>;<br/><b>hence </b><a NAME="E3:86_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:86"/>
<span class="p1">(<span class="default"><a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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