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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<font color="Olive" title="b3">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 holds <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b1">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">k</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 implies (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">and </span><br/><a NAME="E2:69"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<span class="lab"><font color="Green" title="E58">A3</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">m</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D12" target="_self" onmouseover="rs('idea_1/D12')" onmouseout="rh()">Def12</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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="c3">m</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>
;<br/>
<a NAME="E4:69"/>
<font color="Maroon" title="c2">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="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="idea_1.html#D15" target="_self" onmouseover="rs('idea_1/D15')" onmouseout="rh()">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:69"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a>, <a class="ref" href="idea_1.html#T26" target="_self" onmouseover="rs('idea_1/T26')" onmouseout="rh()">Th26</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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