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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:8"/>
 <a href="funcop_1.html#K9">rng</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" target="_self">Inv</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:8"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E4:8"/><i><font color="Green">E9</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" target="_self">Inv</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="pralg_1.html#K6" target="_self">Inv</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="pralg_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E6:8"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="funcop_1.html#K9">rng</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" target="_self">Inv</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E4:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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