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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">d</font>, <font color="Olive" title="b3">d1</font>, <font color="Olive" title="b4">d2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Olive" title="b1">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">d1</font>,<font color="Olive" title="b4">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">d1</font></span>)</span>,<font color="Olive" title="b4">d2</font></span><br/><b>let </b><font color="Maroon" title="c2">b</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon" title="c2">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">d1</font></span>)</span>,<font color="Olive" title="b3">d2</font></span><br/><b>let </b><font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font></span>)</span>,<font color="Maroon" title="c5">d2</font></span><br/>









<b>set </b><font color="Maroon" title="c6">F</font> = <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span>;<br/>
<a NAME="E1:62"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T43" title="AFINSQ_1:th.43">AFINSQ_1:43</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E3:62"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E4:62"/><i><font color="Green" title="E40">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 holds <br/><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:62"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#D3" target="_self" title="STIRL2_1:def.3">Def3</a></i>;<br/>
<a NAME="E6:62"/>
( <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp; <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d1</font> &amp; <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d2</font> )
 
<b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="afinsq_1.html#T43" title="AFINSQ_1:th.43">AFINSQ_1:43</a></i>;<br/>
<b>hence </b><a NAME="E7:62"/>
<font color="Maroon" title="c2">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1"><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font></span><a href="stirl2_1.html#K6" title="STIRL2_1:func.6">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">d1</font></span>)</span>,<font color="Maroon" title="c5">d2</font>
 <b>by </b><i><a class="txt" href="#E3:62"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E5:62"><i><font color="Green" title="E41">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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