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<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E2:71_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E3:71_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#T52" target="_self">Th52</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="txt" href="supinf_2.html#E3:71_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>assume </b><a NAME="E5:71_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_2.html#D14" target="_self">SUPINF_2:def 14</a><br/>

<b>thus </b><a NAME="E6:71_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


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