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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">x4</font>, <font color="Olive" title="b5">x5</font>, <font color="Olive" title="b6">x6</font>, <font color="Olive" title="b7">x7</font>, <font color="Olive" title="b8">x8</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b9">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b9">s</font> is 8 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  &amp; <font color="Olive" title="b9">s</font> <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="Olive" title="b1">x1</font> &amp; <font color="Olive" title="b9">s</font> <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="Olive" title="b2">x2</font> &amp; <font color="Olive" title="b9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x3</font> &amp; <font color="Olive" title="b9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x4</font> &amp; <font color="Olive" title="b9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">x5</font> &amp; <font color="Olive" title="b9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">x6</font> &amp; <font color="Olive" title="b9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">x7</font> &amp; <font color="Olive" title="b9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">x8</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">x3</font>, <font color="Maroon" title="c5">x4</font>, <font color="Maroon" title="c6">x5</font>, <font color="Maroon" title="c7">x6</font>, <font color="Maroon" title="c8">x7</font>, <font color="Maroon" title="c9">x8</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <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">  ex <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">s</font> is 8 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  &amp; <font color="Olive" title="b1">s</font> <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="c2">x1</font> &amp; <font color="Olive" title="b1">s</font> <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="c3">x2</font> &amp; <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x3</font> &amp; <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x4</font> &amp; <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x5</font> &amp; <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x6</font> &amp; <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x7</font> &amp; <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x8</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c10">a1</font> = <span class="p1"><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c5">x4</font></span><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">a2</font> = <span class="p1"><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x5</font>,<font color="Maroon" title="c7">x6</font>,<font color="Maroon" title="c8">x7</font>,<font color="Maroon" title="c9">x8</font></span><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">*&gt;</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c12">b1</font> = <span class="p1"><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c5">x4</font></span><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">*&gt;</a></span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x5</font>,<font color="Maroon" title="c7">x6</font>,<font color="Maroon" title="c8">x7</font>,<font color="Maroon" title="c9">x8</font></span><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">*&gt;</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"> ( <font color="Maroon" title="c12">b1</font> is 8 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  &amp; <font color="Maroon" title="c12">b1</font> <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="c2">x1</font> &amp; <font color="Maroon" title="c12">b1</font> <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="c3">x2</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x3</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x4</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x5</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x6</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x7</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x8</font> )</span><br/>

<a NAME="E1:38"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c5">x4</font></span><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  ...  &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c5">x4</font></span><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T154" onmouseover="rs('finseq_3/T154')" onmouseout="rh()">FINSEQ_3:154</a></span>;<br/>
<a NAME="E2:38"/>
<font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x5</font>,<font color="Maroon" title="c7">x6</font>,<font color="Maroon" title="c8">x7</font>,<font color="Maroon" title="c9">x8</font></span><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  ...  &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x5</font>,<font color="Maroon" title="c7">x6</font>,<font color="Maroon" title="c8">x7</font>,<font color="Maroon" title="c9">x8</font></span><a href="finseq_4.html#K9" title="FINSEQ_4:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T155" onmouseover="rs('finseq_3/T155')" onmouseout="rh()">FINSEQ_3:155</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:38"/>
( <font color="Maroon" title="c12">b1</font> is 8 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  &amp; <font color="Maroon" title="c12">b1</font> <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="c2">x1</font> &amp; <font color="Maroon" title="c12">b1</font> <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="c3">x2</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x3</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x4</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x5</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x6</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x7</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x8</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E14">A1</font></span></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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