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<a NAME="E1:57_2_1"/>
 <a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="aofa_i00.html#D22" target="_self" title="AOFA_I00:def.22">INT'iwa</a></i>;<br/>
<a NAME="E2:57_2_1"/><b>then </b>
 <a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="aofa_i00.html#D22" target="_self" title="AOFA_I00:def.22">INT'iwa</a></i>;<br/>
<a NAME="E3:57_2_1"/><b>then </b><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <span class="p1">(<span class="default"><a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <span class="p2">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="aofa_i00.html#D22" target="_self" title="AOFA_I00:def.22">INT'iwa</a></i>;<br/>
<a NAME="E4:57_2_1"/><i><font color="Green" title="E36">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="c5">t</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <span class="p3">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="aofa_i00.html#K8" title="AOFA_I00:func.8">-</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <span class="p2">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <span class="p3">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="aofa_i00.html#K8" title="AOFA_I00:func.8">-</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> )</span><br/><b>assume </b><a NAME="E1:57_2_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> 
 ; <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="c5">t</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <span class="p3">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="aofa_i00.html#K8" title="AOFA_I00:func.8">-</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font></span><br/><b>then reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">a</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <span class="p3">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <span class="p3">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:57_2_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="aofa_i00.html#K8" title="AOFA_I00:func.8">-</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:57_2_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</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>hence </b><a NAME="E6:57_2_1"/>
<font color="Maroon" title="c5">t</font> <a href="valued_1.html#K13" title="VALUED_1:func.13">-</a> <font color="Maroon" title="c6">i</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="aofa_i00.html#E3:57_2_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="aofa_i00.html#E4:57_2_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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