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<div class="add">

<a NAME="E1:2_1_2"/>
for <font color="Olive" title="b1">IT</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> ( <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font> iff for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b1">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">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" title="c3">IT</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">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" title="E2">A2</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font>
 ;<br/><a NAME="E2:2_1_2_1_1"/><b>then </b><i><font color="Green" title="E3">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><a NAME="E3:2_1_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="diff_1.html#E2:2_1_2_1_1"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/><a NAME="E4:2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="measure6.html#K6" title="MEASURE6:func.6">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="diff_1.html#D1" target="_self" title="DIFF_1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:2_1_2_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="diff_1.html#E1:2_1_2_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="diff_1.html#D1" target="_self" title="DIFF_1:def.1">Def1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:2_1_2_1"/><i><font color="Green" title="E2">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">h</font></span>)</span>
 ;<br/>

<a NAME="E3:2_1_2_1"/><i><font color="Green" title="E3">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="measure6.html#K6" title="MEASURE6:func.6">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="diff_1.html#D1" target="_self" title="DIFF_1:def.1">Def1</a></i>;<br/>
<a NAME="E4:2_1_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:2_1_2_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="measure6.html#T60" title="MEASURE6:th.60">MEASURE6:60</a>, <a class="txt" href="diff_1.html#E3:2_1_2_1"><i><font color="Green" title="E3">A5</font></i></a></i>;<br/>
<a NAME="E6:2_1_2_1"/><b>then </b><i><font color="Green" title="E4">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:2_1_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">IT</font> holds <br/><font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<a NAME="E1:2_1_2_1_2"/><i><font color="Green" title="E5">A7</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="diff_1.html#E2:2_1_2_1"><i><font color="Green" title="E2">A4</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:2_1_2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">IT</font>
 ;<br/>

<a NAME="E3:2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="measure6.html#K6" title="MEASURE6:func.6">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="diff_1.html#D1" target="_self" title="DIFF_1:def.1">Def1</a>, <a class="txt" href="diff_1.html#E6:2_1_2_1"><i><font color="Green" title="E4">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:2_1_2_1_2"/>
<font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="diff_1.html#D1" target="_self" title="DIFF_1:def.1">Def1</a>, <a class="txt" href="diff_1.html#E1:2_1_2_1_2"><i><font color="Green" title="E5">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:2_1_2_1"/>
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font>
 <b>by </b><i><a class="txt" href="diff_1.html#E6:2_1_2_1"><i><font color="Green" title="E4">A6</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">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" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diff_1.html#K1" title="DIFF_1:func.1">Shift</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">h</font> iff for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">h</font></span>)</span> )
 ;<br/>


</div>
