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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Si</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( <font color="Olive" title="b2">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">Si</font> iff <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Olive" title="b1">Si</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Si</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( <font color="Olive" title="b1">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font> iff <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">Si</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font> iff <font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">Si</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:4"/>
( <font color="Maroon" title="c3">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font> implies <font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">Si</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">Si</font> implies <font color="Maroon" title="c3">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:4_1"/>
<font color="Maroon" title="c3">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">Si</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font> ;<br/>
<a NAME="E3:4_1"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T1" target="_self" title="PROB_4:th.1">Th1</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:4_1"/>
<font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">Si</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:4"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">Si</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></span>;<br/>
<a NAME="E4:4"/>
<font color="Maroon" title="c4">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E4">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:4"/>
<font color="Maroon" title="c3">f</font> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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