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<b>set </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font>;<br/>
<a NAME="E1:244_1_1"/><i><font color="Green" title="E118">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="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T131" target="_self" title="ABCMIZ_1:th.131">ThIV</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_1.html#D58" target="_self" title="ABCMIZ_1:def.58">ABCMIZ_1:def 58</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span>
<div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> implies  ex <font color="Olive" title="b1">y</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> )</span><br/><b>assume </b><a NAME="E1:244_1_1_1"/><i><font color="Green" title="E119">A0</font></i>: 
<font color="Maroon" title="c4">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="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span><br/><b>take </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span><br/><b>thus </b><a NAME="E2:244_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="#E1:244_1_1_1"><i><font color="Green" title="E119">A0</font></i></a>, <a class="txt" href="#E1:244_1_1"><i><font color="Green" title="E118">A1</font></i></a>, <a class="ref" href="abcmiz_1.html#T131" target="_self" title="ABCMIZ_1:th.131">ThIV</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font>, <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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<b>assume </b><a NAME="E3:244_1_1"/><i><font color="Green" title="E119">A2</font></i>: 
( <font color="Maroon" title="c4">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="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c5">y</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="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> ;<br/>
<a NAME="E5:244_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> )
 
<b>by </b><i><a class="txt" href="#E1:244_1_1"><i><font color="Green" title="E118">A1</font></i></a>, <a class="txt" href="#E3:244_1_1"><i><font color="Green" title="E119">A2</font></i></a>, <a class="ref" href="abcmiz_1.html#T131" target="_self" title="ABCMIZ_1:th.131">ThIV</a></i>;<br/>
<b>hence </b><a NAME="E6:244_1_1"/>
(  not <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="abcmiz_1.html#K54" title="ABCMIZ_1:func.54">idval</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T50" target="_self" title="ABCMIZ_1:th.50">ThT0</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
