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<b>let </b><font color="Maroon" title="c1">FT1</font>, <font color="Maroon" title="c2">FT2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">h</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">FT1</font>,<font color="Maroon" title="c2">FT2</font>;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="fintopo5.html#R1" title="FINTOPO5:pred.1">is_homeomorphism</a> 
 ;<br/>

<a NAME="E2:4"/><b>then </b><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c3">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">h</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT1</font> holds  <font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">FT2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="fintopo5.html#D1" target="_self" title="FINTOPO5:def.1">Def1</a></i>;<br/>
<a NAME="E3:4"/><b>then </b><i><font color="Green" title="E6">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">FT2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">g2</font> = <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">FT2</font>,<font color="Maroon" title="c1">FT1</font> <b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c4">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="funct_1.html#T62" title="FUNCT_1:th.62">FUNCT_1:62</a></i>;<br/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g2</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h</font>
<b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">FT1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E7:4"/><b>then </b><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c4">g2</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E8:4"/>
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">FT2</font> holds  <font color="Maroon" title="c4">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">FT1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">FT2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT1</font> ;<br/>
<a NAME="E2:4_2"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="fintopo5.html#E3:4"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>
<a NAME="E3:4_2"/>
<font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">FT2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="fintopo5.html#D1" target="_self" title="FINTOPO5:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:4_2"/>
<font color="Maroon" title="c4">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">FT1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="fintopo5.html#E2:4_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="fintopo5.html#T1" target="_self" title="FINTOPO5:th.1">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:4"/><b>then </b>
<font color="Maroon" title="c4">g2</font> <a href="fintopo5.html#R1" title="FINTOPO5:pred.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="fintopo5.html#E7:4"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="fintopo5.html#D1" target="_self" title="FINTOPO5:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:4"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">FT2</font>,<font color="Maroon" title="c1">FT1</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive" title="b1">g</font> <a href="fintopo5.html#R1" title="FINTOPO5:pred.1">is_homeomorphism</a>  )
 ;<br/>


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