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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#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>, <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="c1">D</font>  st  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Olive" title="b2">d2</font> holds <br/><font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">d1</font>, <font color="Maroon" title="c3">d2</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</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#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c3">d2</font> implies <font color="Maroon" title="c2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:1"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c3">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="c2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">Xd1</font> =  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c2">d1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Xd2</font> =  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c3">d2</font>;<br/>
<a NAME="E2:1"/><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#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c2">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="c2">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="E3:1"/><span class="lab"><font color="Green" title="E6">A3</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#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c3">d2</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">d2</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="E4:1"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">d1</font> holds <br/><font color="Maroon" title="c2">d1</font> <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> <font color="Maroon" title="c3">d2</font> <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="1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">d1</font> implies <font color="Maroon" title="c2">d1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">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"> <font color="Maroon" title="c2">d1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/>

<a NAME="E2:1_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <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="c2">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E3:1_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 
;<br/>
<a NAME="E4:1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">d1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c2">d1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="afinsq_1.html#D9" onmouseover="rs('afinsq_1/D9')" onmouseout="rh()">AFINSQ_1:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:1_1"/>
<font color="Maroon" title="c2">d1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="afinsq_1.html#D9" onmouseover="rs('afinsq_1/D9')" onmouseout="rh()">AFINSQ_1:def 9</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="E5:1"/>
<font color="Maroon" title="c2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="afinsq_1.html#T9" onmouseover="rs('afinsq_1/T9')" onmouseout="rh()">AFINSQ_1:9</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>
