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<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">C0</font> be   <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>, <font color="Maroon">KXP</font> be   <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>, <font color="Maroon">KXN</font> be   <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>, <font color="Maroon">KYP</font> be   <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>, <font color="Maroon">KYN</font> be   <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/><b>let </b><font color="Maroon">O</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">I</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">C0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 </span>}</span>  &amp; <font color="Maroon">KXP</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</font> is   <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> : ( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) </span>}</span>  &amp; <font color="Maroon">KXN</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</font> is   <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> : ( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">q2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) </span>}</span>  &amp; <font color="Maroon">KYP</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q3</font> where <font color="Olive">q3</font> is   <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> : ( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">q3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">q3</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) </span>}</span>  &amp; <font color="Maroon">KYN</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q4</font> where <font color="Olive">q4</font> is   <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> : ( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q4</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">q4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">q4</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) </span>}</span>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">KXN</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">KXP</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">KYP</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">KYN</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C0</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:14"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g2</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g2</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:14"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/>


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