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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>







<span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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/>
<span class="kw">set </span><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/>
<span class="kw">reconsider </span><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/>
<span class="kw">set </span><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"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T9" target="_self" title="LPSPACE1:th.9">Th9</a></span>;<br/>
<a NAME="E5:28"/><span class="lab"><font color="Green" title="E20">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="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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T6" target="_self" title="LPSPACE1:th.6">Th6</a></span>;<br/>
<a NAME="E6:28"/><span class="lab"><font color="Green" title="E21">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="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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T6" target="_self" title="LPSPACE1:th.6">Th6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E22">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">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/><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D4" target="_self" title="LPSPACE1:def.4">Def4</a></span>;<br/><span class="kw">assume </span><a NAME="E2:28_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c10">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="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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c10">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"><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/><a NAME="E4:28_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D4" target="_self" title="LPSPACE1:def.4">Def4</a></span>;<br/><a NAME="E5:28_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">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> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T9" target="_self" title="LPSPACE1:th.9">Th9</a></span>;<br/><a NAME="E7:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">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"><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><a NAME="E8:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><a NAME="E9:28_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E10:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">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> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T9" target="_self" title="LPSPACE1:th.9">Th9</a></span>;<br/><a NAME="E11:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">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"><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><a NAME="E12:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="lpspace1.html#T6" target="_self" title="LPSPACE1:th.6">Th6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D4" title="RFUNCT_3:def.4">RFUNCT_3:def 4</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="lpspace1.html#T9" target="_self" title="LPSPACE1:th.9">Th9</a></span>
; <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>
<a NAME="E8:28"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="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 class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T9" target="_self" title="LPSPACE1:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:28"/>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>; <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>
