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<b>let </b><font color="Maroon" title="c1">Z</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">X</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"> for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> implies <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> )</span><br/>







<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span></span><br/>

<a NAME="E2:36"/><i><font color="Green" title="E18">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c5">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"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> or <font color="Maroon" title="c5">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"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> )</span><br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c5">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"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">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"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span></span><br/>

<a NAME="E2:36_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T78" target="_self" title="PARTFUN1:th.78">Th78</a></i>;<br/>
<a NAME="E3:36_1"/><b>then </b>
( <font color="Maroon" title="c5">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> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="txt" href="#E1:36"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E1:36_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="partfun1.html#T78" target="_self" title="PARTFUN1:th.78">Th78</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1"/>
<font color="Maroon" title="c5">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"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 <b>by </b><i><a class="ref" href="partfun1.html#T78" target="_self" title="PARTFUN1:th.78">Th78</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="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</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="c5">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"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> implies <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><b>assume </b><a NAME="E1:36_2"/>
<font color="Maroon" title="c5">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"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><a NAME="E2:36_2"/><b>then </b>
( <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c5">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"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E2:36"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="partfun1.html#T80" target="_self" title="PARTFUN1:th.80">Th80</a></i>;<br/><b>hence </b><a NAME="E3:36_2"/>
<span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T80" target="_self" title="PARTFUN1:th.80">Th80</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="E4:36"/>
<span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Z</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E2:36"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</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>
