<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:49"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<a NAME="E2:49"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:49_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:49_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E1:49_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:49_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E2:49_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:49_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:49_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="tarski.html#R1">c=</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>
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T39" target="_self">Th39</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E7:49_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:49_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jgraph_3.html#E3:49_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jgraph_3.html#E5:49_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jgraph_3.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E58</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_3.html#E1:49"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
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> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="jgraph_3.html#T39" target="_self">Th39</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E4:49"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <a href="pscomp_1.html#K13">proj1</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E5:49"/>
 <a href="relset_1.html#K5">rng</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E6:49"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E4:49"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:49"/>
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="txt" href="jgraph_3.html#E3:49"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>


</div>
