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<span class="kw">let </span><font color="Maroon" title="c1">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>  st <font color="Maroon" title="c1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> holds <br/> for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<font color="Olive" title="b2">n</font>,<font color="Maroon" title="c1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">v1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v1</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> implies  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">v1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">v1</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> ($1,<font color="Maroon" title="c1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">v1</font>;<br/>
<span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c1">v</font>) = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#D3" onmouseover="rs('binom/D3')" onmouseout="rh()">BINOM:def 3</a></span>
<br/>.= 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">v1</font>

;<br/>
<a NAME="E3:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">X1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E4:30"/><span class="lab"><font color="Green" title="E18">X2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E19">X22</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">v</font>) = 
<font color="Maroon" title="c1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">v</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#D3" onmouseover="rs('binom/D3')" onmouseout="rh()">BINOM:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c2">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E19">X22</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">v1</font>

;<br/>
<span class="kw">hence </span><a NAME="E3:30_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:30"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E17">X1</font></span></a>, <a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E18">X2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:30"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">v1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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