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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">D</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"> <span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  <a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">F</font> = <span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>;<br/>
<a NAME="E1:30"/><span class="lab"><font color="Green" title="E19">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="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font>
 
<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/>
<div><span class="lab"><font color="Green" title="E20">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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="c1">D</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="c1">D</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font></span><br/><div><span class="lab"><font color="Green" title="E21">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</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="c4">d</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">G</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font> )</span><br/><span class="kw">assume </span><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c4">d</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">G</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></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"><font color="Maroon" title="c3">G</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c3">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font>

;<br/> <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="E2:30_1"/>
<span class="p1">(<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#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">D</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">G</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/><a NAME="E3:30_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></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="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E19">A1</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="E4:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">G</font> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:30_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D4" target="_self" title="RFUNCT_3:def.4">Def4</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="E3:30"/>
 <a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T14" target="_self" title="RFUNCT_3:th.14">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:30"/>
<span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  <a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="binop_1.html#T5" title="BINOP_1:th.5">BINOP_1:5</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>
