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<b>let </b><font color="Maroon">S</font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">T</font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font>
 ;<br/>

<b>assume </b><a NAME="E3:19"/>
not <font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a>  ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:19"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:19"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E7"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="topreal2.html#E8"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<a NAME="E8:19"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E9:19"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal2.html#E6"><i><font color="Green">Lemma36</font></i></a>, <a class="txt" href="topreal2.html#E9"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>


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