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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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/>









<span class="kw">set </span><font color="Maroon" title="c6">F</font> = <span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&gt;</a></span>;<br/>
<a NAME="E1:66"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T39" title="AFINSQ_1:th.39">AFINSQ_1:39</a></span>;<br/>
<a NAME="E2:66"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T39" title="AFINSQ_1:th.39">AFINSQ_1:39</a></span>;<br/>
<a NAME="E3:66"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T39" title="AFINSQ_1:th.39">AFINSQ_1:39</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:66"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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>
 <span class="kw">and </span><br/><a NAME="E6:66"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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>)
 <span class="kw">and </span><br/><a NAME="E7:66"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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#K21" title="BINOP_2:func.21">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D8" target="_self" title="AFINSQ_2:def.8">Def9</a></span>;<br/>
<a NAME="E8:66"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E43">A4</font></span></a></span>;<br/>
<a NAME="E9:66"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E43">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:66"/>
<font color="Maroon" title="c2">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1"><a href="afinsq_1.html#K7" title="AFINSQ_1:func.7">&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="afinsq_1.html#K7" title="AFINSQ_1:func.7">%&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E8:66"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="afinsq_1.html#T39" title="AFINSQ_1:th.39">AFINSQ_1:39</a></span>; <a class="txt" 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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