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<span class="kw">let </span><font color="Maroon" title="c3">K1</font> be    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</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">  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K1</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">C1</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K1</font>;<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>$1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K1</font>;<br/>
<a NAME="E1:71"/>
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E2:71"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>;<br/>
<a NAME="E3:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<div><span class="lab"><font color="Green" title="E29">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">now </span> <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>  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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:71_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">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="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/><span class="lab"><font color="Green" title="E31">A4</font></span>:  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K1</font> = 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>
<br/>.= 
1

;<br/><a NAME="E3:71_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>;<br/> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">K1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c5">n</font>) = 
<a href="binop_2.html#K15" title="BINOP_2:func.15">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c5">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D15" onmouseover="rs('binop_2/D15')" onmouseout="rh()">BINOP_2:def 15</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:71_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="binop_1.html#T17" onmouseover="rs('binop_1/T17')" onmouseout="rh()">BINOP_1:17</a></span>; <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="E5:71"/>
 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="#E3:71"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E29">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:71"/>
 <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K1</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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