<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">Y1</font>, <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">Y2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c3">X2</font>;<br/><b>let </b><font color="Maroon" title="c6">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Y1</font>,<font color="Maroon" title="c4">Y2</font>;<br/>





<a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y1</font> &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">f</font> <a href="lang1.html#K16" title="LANG1:func.16">*</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</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="c6">g</font> <a href="lang1.html#K16" title="LANG1:func.16">*</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:5"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">g</font>
 ;<br/>

<a NAME="E3:5"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/>
<a NAME="E4:5"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c1">X1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="txt" href="algspec1.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="finseq_1.html#T83" title="FINSEQ_1:th.83">FINSEQ_1:83</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">X1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  ;<br/><a NAME="E3:5_1"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 ;<br/><a NAME="E4:5_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="lang1.html#K16" title="LANG1:func.16">*</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c5">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">p</font> )
 <b>by </b><i><a class="txt" href="algspec1.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="algspec1.html#E2:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="algspec1.html#E4:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="algspec1.html#T2" target="_self" title="ALGSPEC1:th.2">Th2</a>, <a class="ref" href="lang1.html#D14" title="LANG1:def.14">LANG1:def 14</a></i>;<br/><b>hence </b><a NAME="E5:5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="lang1.html#K16" title="LANG1:func.16">*</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="lang1.html#K16" title="LANG1:func.16">*</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="ref" href="lang1.html#D14" title="LANG1:def.14">LANG1:def 14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:5"/>
<font color="Maroon" title="c5">f</font> <a href="lang1.html#K16" title="LANG1:func.16">*</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">g</font> <a href="lang1.html#K16" title="LANG1:func.16">*</a> 
 <b>by </b><i><a class="txt" href="algspec1.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="algspec1.html#E4:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>


</div>
