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<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">lk</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Key1</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">lk</font>,6, <a href="numbers.html#K5">NAT</a> , <font color="Maroon">Key2</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">lk</font>,6, <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">ks1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">ks2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">ke1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">ke2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:78"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">lk</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:78"/><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 is <a href="int_2.html#V1">prime</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 )
 <b>and </b><br/><a NAME="E3:78"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E4:78"/><i><font color="Green">E39</font></i>: 
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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> holds <br/>( <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 )
 <b>and </b><br/><a NAME="E5:78"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">ks1</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ks1</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ks1</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ks1</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ks2</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">ks1</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">ks2</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">ks1</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">ks2</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">ks1</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">ks2</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">ks1</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E6:78"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ke2</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">ke2</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">ke2</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">ke2</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">ke2</font> <a href="wsierp_1.html#K3">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 5 &amp; <font color="Maroon">ke2</font> <a href="wsierp_1.html#K3">.</a> 6 <a href="hidden.html#R1">=</a> <font color="Maroon">ke1</font> <a href="wsierp_1.html#K3">.</a> 6 )
 ;<br/>

<a NAME="E7:78"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:78"/><b>then </b>
<span class="p1">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m1</font> = <span class="p1">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<i><font color="Green">E44</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m1</font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <font color="Maroon">m</font>,<font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font>
<b>by </b><i/>
;<br/>
<a NAME="E11:78"/>
( <font color="Maroon">m1</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <font color="Maroon">m</font>,<font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 1 &amp; <font color="Maroon">m1</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <font color="Maroon">m</font>,<font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 2 &amp; <font color="Maroon">m1</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <font color="Maroon">m</font>,<font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 3 &amp; <font color="Maroon">m1</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <font color="Maroon">m</font>,<font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 4 )
 
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E12:78"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">m1</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m1</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m1</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m1</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:78"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_2"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1"><b>suppose </b></a><a NAME="E1:78_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:78_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:78_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:78_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="idea_1.html#E6:78"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <a href="xboole_0.html#K1">{}</a> ,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4">compose</a> <a href="xboole_0.html#K1">{}</a> ,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="idea_1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4">compose</a> <a href="xboole_0.html#K1">{}</a> ,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="funct_7.html#T41">FUNCT_7:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="funct_7.html#T41">FUNCT_7:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="idea_1.html#T4" target="_self">Th4</a>, , <a class="txt" href="idea_1.html#E3:78_2_1"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="idea_1.html#T1" target="_self">Th1</a>, , <a class="txt" href="idea_1.html#E1:78_2_1"><i><font color="Green">E48</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_2"><b>suppose </b></a><a NAME="E1:78_2_2"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:78_2_2"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font> is    <a href="funct_7.html#M1">FuncSequence</a> of  <a href="idea_1.html#K14" target="_self">MESSAGES</a> , <a href="idea_1.html#K14" target="_self">MESSAGES</a>  &amp;  <a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font> is    <a href="funct_7.html#M1">FuncSequence</a> of  <a href="idea_1.html#K14" target="_self">MESSAGES</a> , <a href="idea_1.html#K14" target="_self">MESSAGES</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E3:78_2_2"/>
 <a href="funct_7.html#K4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a>  is   <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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:78_2_2_1"/><i><font color="Green">E53</font></i>: 
(  <a href="funct_7.html#K6">firstdom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  &amp;  <a href="funct_7.html#K7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  )
 
<b>by </b><i>, <a class="ref" href="funct_7.html#D9">FUNCT_7:def 9</a></i>;<br/>
<a NAME="E2:78_2_2_1"/><b>then </b>
(  <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_7.html#D6">FUNCT_7:def 6</a></i>;<br/>
<a NAME="E3:78_2_2_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_7.html#K7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T61">FUNCT_7:61</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:78_2_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="idea_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:78_2_2_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i>, <a class="ref" href="idea_1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_7.html#T64">FUNCT_7:64</a></i>;<br/>
<b>hence </b><a NAME="E6:78_2_2_1"/>
 <a href="funct_7.html#K4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a>  is   <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> 
 <b>by </b><i><a class="ref" href="idea_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">PF</font> =  <a href="funct_7.html#K4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a>  as   <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/><a NAME="E5:78_2_2"/>
 <a href="funct_7.html#K4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a>  is   <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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:78_2_2_2"/><i><font color="Green">E56</font></i>: 
(  <a href="funct_7.html#K6">firstdom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  &amp;  <a href="funct_7.html#K7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  )
 
<b>by </b><i>, <a class="ref" href="funct_7.html#D9">FUNCT_7:def 9</a></i>;<br/>
<a NAME="E2:78_2_2_2"/><b>then </b>
(  <a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_7.html#D6">FUNCT_7:def 6</a></i>;<br/>
<a NAME="E3:78_2_2_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_7.html#K7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T61">FUNCT_7:61</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:78_2_2_2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="idea_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:78_2_2_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i>, <a class="ref" href="idea_1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_7.html#T64">FUNCT_7:64</a></i>;<br/>
<b>hence </b><a NAME="E6:78_2_2_2"/>
 <a href="funct_7.html#K4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a>  is   <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> 
 <b>by </b><i><a class="ref" href="idea_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">QF</font> =  <a href="funct_7.html#K4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a>  as   <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/><a NAME="E7:78_2_2"/><i><font color="Green">E59</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">PF</font> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">QF</font> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E8:78_2_2"/>
<font color="Maroon">PF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">m2</font> = <font color="Maroon">PF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E10:78_2_2"/><i><font color="Green">E60</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 &amp; <font color="Maroon">m2</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m2</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m2</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m2</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="idea_1.html#E11:78_2_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="idea_1.html#T4" target="_self">Th4</a>, , </i>;<br/><a NAME="E11:78_2_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">m2</font> <a href="hidden.html#R2">in</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E12:78_2_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">QF</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">PF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2_3"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon">QF</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">PF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">QF</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">PF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font>
<b>by </b><i><a class="ref" href="idea_1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_7.html#T50">FUNCT_7:50</a></i>
<br/>.= 
<font color="Maroon">m1</font>
<b>by </b><i><a class="txt" href="idea_1.html#E10:78_2_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="idea_1.html#E11:78_2_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="idea_1.html#T2" target="_self">Th2</a>, <a class="ref" href="idea_1.html#T4" target="_self">Th4</a>, , <a class="txt" href="idea_1.html#E7:78_2_2"><i><font color="Green">E59</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">QF</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">PF</font> <a href="partfun1.html#K1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">QF</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">PF</font> <a href="partfun1.html#K1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">QF</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">PF</font> <a href="partfun1.html#K1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">QF</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">PF</font> <a href="partfun1.html#K1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">QF</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">PF</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">QF</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">ke2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">ke1</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font>
<b>by </b><i><a class="txt" href="idea_1.html#E11:78_2_2"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="idea_1.html#D2" target="_self">Def2</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" target="_self">IDEA_QS</a> <font color="Maroon">ks2</font>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K19" target="_self">IDEA_PS</a> <font color="Maroon">ks1</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="idea_1.html#E11:78_2_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="idea_1.html#T1" target="_self">Th1</a>, , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
