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

<span class="kw">let </span><font color="Maroon" title="c1">A</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">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b3">h</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">A</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"> <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">j</font> = <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">k</font> = <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">j1</font> = <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">k1</font> = <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font>;<br/>
<a NAME="E1:21"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span></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">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T7" target="_self" title="LPSPACE1:th.7">Th7</a></span>;<br/>
<a NAME="E2:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>
 
<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/>
<a NAME="E3:21"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span>)</span></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">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="lpspace1.html#T7" target="_self" title="LPSPACE1:th.7">Th7</a></span>;<br/>
<a NAME="E4:21"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T7" target="_self" title="LPSPACE1:th.7">Th7</a></span>;<br/>
<a NAME="E5:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T7" target="_self" title="LPSPACE1:th.7">Th7</a></span>;<br/>
<a NAME="E6:21"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><a NAME="E2:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="lpspace1.html#T7" target="_self" title="LPSPACE1:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="lpspace1.html#T7" target="_self" title="LPSPACE1:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="lpspace1.html#T7" target="_self" title="LPSPACE1:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="lpspace1.html#T7" target="_self" title="LPSPACE1:th.7">Th7</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">hence </span><a NAME="E8:21"/>
<span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K3" title="LPSPACE1:func.3">multpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</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>
