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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  ) or (  ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> ) or (  ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font> ) );<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:14_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font></span>)</span>,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font></span>)</span>,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T42">ENUMSET1:42</a></i>;<br/><a NAME="E2:14_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font></span>)</span>,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> &amp;  <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font></span>)</span>,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> &amp;  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font></span>)</span>,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E3:14_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>, <a href="ami_2.html#K3">SCM-Instr-Loc</a> ] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>, <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font>] )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:14_1_1_1"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font></span>)</span>,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">b</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font></span>)</span>,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E3:14_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<a NAME="E4:14_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:14_1_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 ;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E6:14_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>] &amp; <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:14_1_1"/>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>


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