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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being 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/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</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="Olive" title="b1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Olive" title="b1">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#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">f</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</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="c2">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">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#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">f</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">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></span><br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</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="c2">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">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#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span><br/>







<b>reconsider </b><font color="Maroon" title="c5">h</font> = <span class="p1">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">f</font> as    <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="c2">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">i</font> = <span class="p1">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">g</font> as    <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="c2">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>set </b><font color="Maroon" title="c7">j</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">k</font> = <span class="p1">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> as    <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="c2">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>set </b><font color="Maroon" title="c9">l</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c6">i</font>;<br/>
<a NAME="E4:28"/><i><font color="Green" title="E19">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="lpspace1.html#T9" target="_self" title="LPSPACE1:th.9">Th15</a></i>;<br/>
<a NAME="E5:28"/><i><font color="Green" title="E20">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">i</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="lpspace1.html#T6" target="_self" title="LPSPACE1:th.6">Th10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c6">i</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/><a NAME="E2:28_1"/><b>then </b><i><font color="Green" title="E22">B2</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:28"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E5:28"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/><a NAME="E3:28_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">i</font> )
 <b>by </b><i><a class="txt" href="#E1:28_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="#E5:28"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:28_1"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> )
 <b>by </b><i><a class="ref" href="lpspace1.html#T9" target="_self" title="LPSPACE1:th.9">Th15</a></i>;<br/><a NAME="E5:28_1"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rfunct_3.html#K13" title="RFUNCT_3:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rfunct_3.html#K13" title="RFUNCT_3:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> )
 <b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/><a NAME="E6:28_1"/>
( <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rfunct_3.html#K13" title="RFUNCT_3:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rfunct_3.html#K13" title="RFUNCT_3:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="ref" href="lpspace1.html#D4" target="_self" title="LPSPACE1:def.4">Def4</a></i>;<br/><a NAME="E7:28_1"/><b>then </b><i><font color="Green" title="E24">A12</font></i>: 
( <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> &amp; <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E5:28_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:28_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="lpspace1.html#T6" target="_self" title="LPSPACE1:th.6">Th10</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:28_1"><i><font color="Green" title="E24">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:28_1"><i><font color="Green" title="E22">B2</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" title="RFUNCT_3:def.4">RFUNCT_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="#E1:28_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="#E4:28"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E5:28"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="lpspace1.html#T9" target="_self" title="LPSPACE1:th.9">Th15</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:28"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">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#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K4" title="LPSPACE1:func.4">multrealpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:28"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E5:28"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
