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<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>; <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="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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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="E1:273_1_1"/><i><font color="Green" title="E126">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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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/>

<a NAME="E2:273_1_1"/><b>then </b><i><font color="Green" title="E127">A1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D66" target="_self" title="ABCMIZ_1:def.66">SUBST2</a></i>;<br/>
<a NAME="E3:273_1_1"/><i><font color="Green" title="E128">A2</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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D66" target="_self" title="ABCMIZ_1:def.66">SUBST2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="273_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:273_1_1_1"/>
( <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> <font color="Maroon" title="c2">f</font> or <font color="Maroon" title="c4">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="273_1_1_1_1"><b>suppose </b></a><a NAME="E1:273_1_1_1_1"/><i><font color="Green" title="E129">A3</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> <font color="Maroon" title="c2">f</font>
 ; <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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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/><div class="add"><b>then consider </b><font color="Maroon" title="c5">y</font> being   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a><b> such that </b><br/><a NAME="E3:273_1_1_1_1"/><i><font color="Green" title="E130">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <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="ref" href="abcmiz_1.html#D58" target="_self" title="ABCMIZ_1:def.58">IRR</a></i>;<br/><a NAME="E4:273_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c2">f</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="#E1:273_1_1_1_1"><i><font color="Green" title="E129">A3</font></i></a>, <a class="txt" href="#E3:273_1_1_1_1"><i><font color="Green" title="E130">A4</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>;<br/><a NAME="E5:273_1_1_1_1"/><b>then </b>
( ( <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> <font color="Maroon" title="c3">g</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> ) &amp; ( <font color="Maroon" title="c5">y</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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="#E2:273_1_1"><i><font color="Green" title="E127">A1</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>;<br/><b>hence </b><a NAME="E6:273_1_1_1_1"/>
 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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D58" target="_self" title="ABCMIZ_1:def.58">IRR</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="273_1_1_1_2"><b>suppose </b></a><a NAME="E1:273_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ; <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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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/><div class="add"><a NAME="E2:273_1_1_1_2"/><b>then </b><i><font color="Green" title="E129">A7</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> &amp; <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> <font color="Maroon" title="c3">g</font> )
 <b>by </b><i><a class="txt" href="#E1:273_1_1"><i><font color="Green" title="E126">A0</font></i></a>, <a class="txt" href="#E3:273_1_1"><i><font color="Green" title="E128">A2</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:273_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E2:273_1_1"><i><font color="Green" title="E127">A1</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>;<br/><b>hence </b><a NAME="E4:273_1_1_1_2"/>
 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="c2">f</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c3">g</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>
 <b>by </b><i><a class="txt" href="#E2:273_1_1_1_2"><i><font color="Green" title="E129">A7</font></i></a>, <a class="ref" href="abcmiz_1.html#D58" target="_self" title="ABCMIZ_1:def.58">IRR</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></div><b>end;</b></div>

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