<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <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>  holds <br/> ( ( <font color="Olive" title="b1">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="Olive" title="b2">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) iff <font color="Olive" title="b1">f1</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <font color="Olive" title="b2">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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#K8" title="RELSET_1:func.8">"</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>  ) iff <font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <font color="Maroon" title="c3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )</span><br/>





<span class="kw">thus </span><a NAME="E1:77"/>
( <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#K8" title="RELSET_1:func.8">"</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#K3" title="RFUNCT_1:func.3">/</a> <font color="Maroon" title="c3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <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> 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#K8" title="RELSET_1:func.8">"</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>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:77_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c2">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">and </span><br/><a NAME="E2:77_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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> 
 <span class="kw">and </span><br/><a NAME="E3:77_1"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <font color="Maroon" title="c3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> </span><br/>

<a NAME="E4:77_1"/>
<font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a>  is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:77_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="rfunct_1.html#T70" target="_self" title="RFUNCT_1:th.70">Th70</a></span>;<br/>
<a NAME="E5:77_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="rfunct_1.html#T66" target="_self" title="RFUNCT_1:th.66">Th66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:77_1"/>
<font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <font color="Maroon" title="c3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T47" target="_self" title="RFUNCT_1:th.47">Th47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:77"/>
<font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <font color="Maroon" title="c3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="RELSET_1:func.8">"</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>  )</span><br/>

<a NAME="E3:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T47" target="_self" title="RFUNCT_1:th.47">Th47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:77"/>
<font color="Maroon" title="c2">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T66" target="_self" title="RFUNCT_1:th.66">Th66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>  )</span><br/>

<a NAME="E5:77"/>
<font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a>  is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="rfunct_1.html#T66" target="_self" title="RFUNCT_1:th.66">Th66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:77"/>
( <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T70" target="_self" title="RFUNCT_1:th.70">Th70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
