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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<a NAME="E1:73"/><i><font color="Green">E48</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 
;<br/>
<b>thus </b><a NAME="E2:73"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>thus </b> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T53" target="_self">Th53</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:73"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rfunct_3.html#E1:73_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rfinseq.html#T5">RFINSEQ:5</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T53" target="_self">Th53</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:73_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:73"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rfinseq.html#T5">RFINSEQ:5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:73"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><a NAME="E1:73_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>hence </b> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T53" target="_self">Th53</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:73"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rfunct_3.html#E3:73"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rfinseq.html#T5">RFINSEQ:5</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:73_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rfunct_3.html#T53" target="_self">Th53</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:73"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:73"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rfinseq.html#T5">RFINSEQ:5</a></i>;<br/>


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