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<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> 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="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</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"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</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="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</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"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">h</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</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"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>;<br/>
<a NAME="E1:23"/>
 <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="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</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"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</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="c2">f</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="E2:23"/><span class="kw">then </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="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <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="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E3:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">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="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</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"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span> <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="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="c4">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="c2">f</font> 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="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<span class="p1">(<span class="default"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:23_1"/>
<font color="Maroon" title="c4">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="c2">f</font>
 ; <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="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</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"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">hence </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="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> <span class="p2">(<span class="default"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="lpspace1.html#T6" target="_self" title="LPSPACE1:th.6">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>

;<br/> <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="E5:23"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</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"><a href="lpspace1.html#K5" title="LPSPACE1:func.5">RealPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</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>
