<?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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">V</font> holds <br/> ( ( <font color="Olive" title="b2">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) &amp; ( <font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies ( <font color="Olive" title="b2">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) ) &amp; ( <font color="Olive" title="b2">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) &amp; ( <font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b3">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies ( <font color="Olive" title="b2">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Olive" title="b3">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">V</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">V</font> 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> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b2">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) &amp; ( <font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b2">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies ( <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> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) ) &amp; ( <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> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) &amp; ( <font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies ( <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> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) &amp; ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) ) &amp; ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) &amp; ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) ) )</span><br/>







<span class="kw">thus </span><a NAME="E1:41"/>
( ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) iff <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) iff <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">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="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:41_1"/>
( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:41_1"/>
<font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )</span><br/>

<a NAME="E3:41_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E4:41_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D1" target="_self" title="VFUNCT_1:def.1">Def1</a></span>;<br/>
<a NAME="E5:41_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:41_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E2:41"/>
( ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) iff <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:41_2"/>
( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:41_2"/>
<font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )</span><br/>

<a NAME="E3:41_2"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E4:41_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D2" target="_self" title="VFUNCT_1:def.2">Def2</a></span>;<br/>
<a NAME="E5:41_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:41_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div>
