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<div class="add">

<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>for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E1:62_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:62_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:62_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:62_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:62_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:62_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E4:62_1_1_2"/>
for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
;<br/>
<b>hence </b><a NAME="E5:62_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:62_1_1"/>
 ex <font color="Olive">ff</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 
<b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">ff</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:62_1_1"/><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/>for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> holds <br/><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> ) )
 ;<br/>
<a NAME="E6:62_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:62_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:62_1_1_3"/>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:62_1_1_3"/>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>take </b>
<font color="Maroon">ff</font>
;<br/>

<b>thus </b><a NAME="E8:62_1_1"/>
for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds  <font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i/>;<br/>


</div>
