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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">R</font> be    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/>





<i><font color="Green" title="E30">A1</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">*</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">B</font> = 
<font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">B</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>
<br/>.= 
<font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c3">R</font></span>)</span>
<b>by </b><i><a class="ref" href="relset_2.html#T51" target="_self" title="RELSET_2:th.51">Th51</a></i>
;<br/>
<b>thus </b><a NAME="E2:66"/>
<font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">*</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">B</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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><br/>

<b>assume </b><a NAME="E1:66_2"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<a NAME="E2:66_2"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_2.html#T47" target="_self" title="RELSET_2:th.47">Th47</a></i>;<br/>
<a NAME="E3:66_2"/>
<font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T154" title="RELAT_1:th.154">RELAT_1:154</a></i>;<br/>
<b>hence </b><a NAME="E4:66_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">*</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="relset_2.html#E2:66_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="relset_2.html#E1:66_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="relset_2.html#E1:66"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">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><br/>

<b>assume </b><a NAME="E3:66"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">*</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<a NAME="E4:66"/>
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c3">R</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="c3">R</font>
 
<b>by </b><i><a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a></i>;<br/>
<a NAME="E5:66"/><b>then </b>
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:66"/><b>then </b>
<font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E7:66"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="relset_2.html#E3:66"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="relset_2.html#E1:66"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>


</div>
