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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>assume </b><a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E3</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="relat_1.html#V2">non-empty</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="relat_1.html#V2">non-empty</a> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1"/><i><font color="Green">E4</font></i>: 
( <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>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/><a NAME="E4:1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E3:1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:1_1"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_3.html#T37">CARD_3:37</a></i>;<br/><a NAME="E6:1_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="card_3.html#T37">CARD_3:37</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:1_1"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>4</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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E9:1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="pencil_1.html#E8:1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:1"/>
(  not <font color="Maroon">c<sub>1</sub></font> is <a href="relat_1.html#V2">non-empty</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="relat_1.html#V2">non-empty</a> )
 ;<br/>


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