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<a NAME="E1:2_1_2"/>
for <font color="Olive">IT</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  holds <br/> ( <font color="Olive">IT</font> <a href="hidden.html#R1">=</a>  <a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font> iff for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Olive">IT</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">IT</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_1_2_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>
 ;<br/><a NAME="E2:2_1_2_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><a NAME="E3:2_1_2_1_1"/>
<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"><a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:2_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">h</font></span>)</span> <a href="measure6.html#K6">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:2_1_2_1_1"/>
<font color="Maroon">IT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:2_1_2_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">IT</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font></span>)</span>
 ;<br/>

<a NAME="E3:2_1_2_1"/><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">h</font></span>)</span> <a href="measure6.html#K6">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:2_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:2_1_2_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="measure6.html#T60">MEASURE6:60</a>, </i>;<br/>
<a NAME="E6:2_1_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:2_1_2_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">IT</font> holds <br/><font color="Maroon">IT</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:2_1_2_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">IT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:2_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<a NAME="E3:2_1_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">h</font></span>)</span> <a href="measure6.html#K6">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:2_1_2_1_2"/>
<font color="Maroon">IT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:2_1_2_1"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="diff_1.html#K1" target="_self">Shift</a> <font color="Maroon">f</font>,<font color="Maroon">h</font> iff for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font></span>)</span> )
 ;<br/>


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