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<b>let </b><font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c4">m2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> , <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:54_1_2"/><i><font color="Green" title="E39">A4</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">m1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <span class="p1">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <span class="p2">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E2:54_1_2"/><i><font color="Green" title="E40">A5</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">m2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <span class="p1">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <span class="p2">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E3:54_1_2"/>
<font color="Maroon" title="c3">m1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">m2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:54_1_2_1"/><i><font color="Green" title="E41">A6</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1_2_1_1"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c6">jj</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_1_2_1_1"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c6">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">jj</font> = <font color="Maroon" title="c6">jj</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="idea_1.html#E1:54_1_2_1_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="idea_1.html#E3:54_1_2_1_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">m1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> = 
 <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <span class="p1">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <span class="p2">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c7">jj</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:54_1_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="idea_1.html#E3:54_1_2_1_1"><i><font color="Green" title="E43">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">m2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:54_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="idea_1.html#E3:54_1_2_1_1"><i><font color="Green" title="E43">A8</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:54_1_2_1"/>
<font color="Maroon" title="c3">m1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">m2</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:54_1_2_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:54_1_2"/>
<font color="Maroon" title="c3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m2</font>
 ;<br/>


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