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<div class="add">

<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E2:22_2_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">n</font></span>)</span>,<font color="Olive">a</font> ) )
 <b>from </b><i><a class="ref" href="binom.html#S2" target="_self">BINOM:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E3:22_2_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>thus </b><a NAME="E1:22_2_1_1"/>
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>

<a NAME="E2:22_2_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E3:22_2_1_1_1"/>
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_2_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_2_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">a</font> ) )
 ;<br/>


</div>
