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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/>





<a NAME="E1:29"/><i><font color="Green">E37</font></i>: 
(  <a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font> <a href="mesfunc2.html#R1">is_finite</a>  &amp;  <a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font> <a href="mesfunc2.html#R1">is_finite</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E2:29"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T2">MESFUNC2:2</a></i>;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 ;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
;<br/><b>hence </b><a NAME="E3:29_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:29"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#T2">MESFUNC2:2</a></i>;<br/>
<a NAME="E6:29"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 ;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
;<br/><b>hence </b><a NAME="E3:29_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:29"/>
(  <a href="mesfunc5.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="mesfunc5.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">g</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="mesfunc2.html#T2">MESFUNC2:2</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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