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<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">C</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>thus </b><a NAME="E1:63"/>
( <font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> implies <font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green" title="E41">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 ;<br/>

<a NAME="E2:63_1"/><b>then </b>
<font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a>  is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T70" target="_self" title="RFUNCT_1:th.70">Th70</a></i>;<br/>
<a NAME="E3:63_1"/><b>then </b>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:63_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T66" target="_self" title="RFUNCT_1:th.66">Th66</a></i>;<br/>
<b>hence </b><a NAME="E4:63_1"/>
<font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T47" target="_self" title="RFUNCT_1:th.47">Th47</a></i>;<br/>


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

<b>assume </b><a NAME="E2:63"/>
<font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 ;<br/>

<a NAME="E3:63"/><b>then </b><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T47" target="_self" title="RFUNCT_1:th.47">Th47</a></i>;<br/>
<b>hence </b><a NAME="E4:63"/>
<font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T66" target="_self" title="RFUNCT_1:th.66">Th66</a></i>;<br/>

<a NAME="E5:63"/>
<font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a>  is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E3:63"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="rfunct_1.html#T66" target="_self" title="RFUNCT_1:th.66">Th66</a></i>;<br/>
<b>hence </b><a NAME="E6:63"/>
( <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 <b>by </b><i><a class="ref" href="rfunct_1.html#T70" target="_self" title="RFUNCT_1:th.70">Th70</a></i>;<br/>


</div>
