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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">f</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="Olive" title="b1">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c3">f</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="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:92"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</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="c3">f</font>,<font color="Maroon" title="c1">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="E2:92"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">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="92_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></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> <font color="Maroon" title="c3">f</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:92_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<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> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">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="E2:92_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:92_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E1:92_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="partfun1.html#T78" target="_self" title="PARTFUN1:th.78">Th78</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:92"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T77" target="_self" title="PARTFUN1:th.77">Th77</a></span>;<br/>
<a NAME="E4:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E5:92"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="c3">f</font>,<font color="Maroon" title="c1">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="Olive" title="b1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="partfun1.html#T80" target="_self" title="PARTFUN1:th.80">Th80</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:92"/>
<font color="Maroon" title="c3">f</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="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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