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<div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">y1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:60"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font> )
 ;<br/>

<a NAME="E2:60"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> holds <br/><a href="binop_2.html#K28">diffcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K34">diffreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="complsp2.html#K10" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="complsp2.html#K10" target="_self">.</a> <font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:60_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K28">diffcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binop_2.html#D4">BINOP_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:60_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> implies <a href="binop_2.html#K28">diffcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K34">diffreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="complsp2.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="complsp2.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="complsp2.html#T3" target="_self">Th3</a>, <a class="ref" href="binop_2.html#D10">BINOP_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60"/>
<a href="binop_2.html#K28">diffcomplex</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K34">diffreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="complsp2.html#T3" target="_self">Th3</a>, <a class="txt" href="complsp2.html#E2"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


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