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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="idea_1.html#K14" target="_self">MESSAGES</a> , <a href="idea_1.html#K14" target="_self">MESSAGES</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:62_1_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:62_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:62_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_2_1"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:62_1_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:62_1_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="idea_1.html#E1:62_1_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="idea_1.html#E3:62_1_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:62_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="idea_1.html#E3:62_1_2_1_1"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:62_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="idea_1.html#E3:62_1_2_1_1"><i><font color="Green">E49</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:62_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:62_1_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:62_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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