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<span class="kw">let </span><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</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">d1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D1</font><br/>  for <font color="Olive" title="b2">d2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c2">D2</font>  st <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d2</font> holds <br/> <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Olive" title="b2">d2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">d1</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D1</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">  for <font color="Olive" title="b1">d2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c2">D2</font>  st <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d2</font> holds <br/> <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Olive" title="b1">d2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">d2</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c2">D2</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"> ( <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d2</font> implies  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c4">d2</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d2</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">  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c4">d2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">Xd1</font> =  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Xd2</font> =  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c4">d2</font>;<br/>
<a NAME="E2:3"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D9" onmouseover="rs('afinsq_1/D9')" onmouseout="rh()">AFINSQ_1:def 9</a></span>;<br/>
<a NAME="E3:3"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">d1</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be   <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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">d1</font> implies <span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E2:3_1"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">d1</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"> <span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<a NAME="E3:3_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D9" onmouseover="rs('afinsq_1/D9')" onmouseout="rh()">AFINSQ_1:def 9</a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:3_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D9" onmouseover="rs('afinsq_1/D9')" onmouseout="rh()">AFINSQ_1:def 9</a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:3"/>
 <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">XFS2FS</a> <font color="Maroon" title="c4">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="afinsq_1.html#D9" onmouseover="rs('afinsq_1/D9')" onmouseout="rh()">AFINSQ_1:def 9</a>, <a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">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/>


</div>
