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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:7"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E4:7"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><a NAME="E5:7"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<i><font color="Green">E12</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T37">FUNCT_7:37</a></i>
;<br/>
<a NAME="E3:7_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:7"><i><font color="Green">E8</font></i></a>, <a class="txt" href="yellow17.html#E3:7"><i><font color="Green">E9</font></i></a>, <a class="ref" href="yellow17.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E4:7"><i><font color="Green">E10</font></i></a>, <a class="txt" href="yellow17.html#E1:7_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="yellow17.html#E2:7_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow17.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E6:7"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>hence </b><a NAME="E7:7"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E3:7"><i><font color="Green">E9</font></i></a>, <a class="txt" href="yellow17.html#E4:7"><i><font color="Green">E10</font></i></a>, <a class="txt" href="yellow17.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/>


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