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<span class="kw">set </span><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font>;<br/>
<a NAME="E1:151_1_1"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_2.html#D37" target="_self" title="VALUED_2:def.37">Def37</a></span>;<br/>
<a NAME="E2:151_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="valued_2.html#K2" title="VALUED_2:func.2">C_PFuncs</a> <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font></span>)</span> or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="valued_2.html#K2" title="VALUED_2:func.2">C_PFuncs</a> <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:151_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="valued_2.html#K2" title="VALUED_2:func.2">C_PFuncs</a> <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:151_1_1_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c8">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="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:151_1_1_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:151_1_1_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="valued_1.html#NK8" title="VALUED_1:NK.8">+</a> <font color="Maroon" title="c5">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:151_1_1_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="valued_2.html#D37" target="_self" title="VALUED_2:def.37">Def37</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:151_1_1_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a></span>;<br/>
<a NAME="E7:151_1_1_1"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">y</font> or <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:151_1_1_1_1"/>
(  not <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">y</font> or <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:151_1_1_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E5:151_1_1_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:151_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:151_1_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E3:151_1_1_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="partfun1.html#T4" title="PARTFUN1:th.4">PARTFUN1:4</a></span>;<br/>
<a NAME="E9:151_1_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Y</font> : verum </span> } </span> 
 
;<br/>
<a NAME="E10:151_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E11:151_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:151_1_1_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E5:151_1_1_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="valued_1.html#D2" title="VALUED_1:def.2">VALUED_1:def 2</a></span>;<br/>
<a NAME="E12:151_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:151_1_1_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E7:151_1_1_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:151_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="valued_2.html#K2" title="VALUED_2:func.2">C_PFuncs</a> <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_2.html#D8" target="_self" title="VALUED_2:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:151_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="valued_2.html#K30" title="VALUED_2:func.30">[+]</a> <font color="Maroon" title="c5">c</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="valued_2.html#K2" title="VALUED_2:func.2">C_PFuncs</a> <span class="p2">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:151_1_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
