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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font>;<br/>



















<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E1:20_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T113">FINSEQ_2:113</a></i>;<br/><a NAME="E2:20_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#T87">FINSEQ_2:87</a>, <a class="ref" href="finseq_2.html#T115">FINSEQ_2:115</a></i>;<br/><b>hence </b><a NAME="E3:20_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:20"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>
 ;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:20_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/><a NAME="E6:20_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>14</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>16</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="finseqop.html#E3:20_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="finseqop.html#E5:20_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a>, <a class="ref" href="finseq_2.html#T127">FINSEQ_2:127</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseqop.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:20_2"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E3:20_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="finseqop.html#E5:20_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseqop.html#T11" target="_self">Th11</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E4:20"/><b>then </b><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:20"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="finseqop.html#E2:20"><i><font color="Green">E11</font></i></a>, <a class="txt" href="finseqop.html#E4:20"><i><font color="Green">E12</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:20"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/>


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