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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> $1</span>)</span>;<br/>
<a NAME="E1:42_1_1"/><i><font color="Green" title="E40">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">y</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c8">x</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span> <b>by </b><i><a class="ref" href="topalg_1.html#T48" title="TOPALG_1:th.48">TOPALG_1:48</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">y</font>
;<br/>

<b>thus </b><a NAME="E2:42_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span><b> such that </b><br/><a NAME="E3:42_1_1"/><i><font color="Green" title="E41">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="topalg_5.html#E1:42_1_1"><i><font color="Green" title="E40">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="gr_cy_1.html#K3" title="GR_CY_1:func.3">INT.Group</a> ,<span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c9">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c10">n</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<a NAME="E5:42_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:42_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:42_1_1"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>


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