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<a NAME="E1:35_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
<a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> :  <a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Olive" title="b1">f</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="triang_1.html#D7" target="_self" title="TRIANG_1:def.7">Def7</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is <a href="funct_1.html#V1" title="FUNCT_1:attr.1">Function-like</a> &amp; <font color="Maroon" title="c2">x</font> is <a href="relat_1.html#V1" title="RELAT_1:attr.1">Relation-like</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">n'</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:35_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n'</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c4">f</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:35_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E6:35_1_1"/>
 <a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c4">f</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:35_1_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:35_1_1"/>
( <font color="Maroon" title="c2">x</font> is <a href="funct_1.html#V1" title="FUNCT_1:attr.1">Function-like</a> &amp; <font color="Maroon" title="c2">x</font> is <a href="relat_1.html#V1" title="RELAT_1:attr.1">Relation-like</a> )
 <b>by </b><i><a class="txt" href="triang_1.html#E5:35_1_1"><i><font color="Green" title="E13">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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