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<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>(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default">$1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default">$1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>);<br/>
<a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E22">A1</font></span></a></span>;<br/>
<a NAME="E2:31_1"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 for <font color="Olive" title="b1">j</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">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</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="31_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">j</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="c1">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">j1</font> = <font color="Maroon" title="c1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E1:31_1_1"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">(<font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span></span>)</span>) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_1_1_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E2:31_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T235" onmouseover="rs('xreal_1/T235')" onmouseout="rh()">XREAL_1:235</a></span>;<br/>
<a NAME="E3:31_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span>
 
;<br/>
<a NAME="E4:31_1_1_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E5:31_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31_1_1_1"/>
<span class="p1">(<span class="default">(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">(<font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span></span>)</span>) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T14" target="_self" onmouseover="rs('flexary1/T14')" onmouseout="rh()">Th14</a>, <a class="txt" href="#E1:31_1_1_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E3:31_1_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></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="E2:31_1_1"/>
<span class="p1">(<span class="default">(<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">(<font color="Maroon">F<sub>2</sub></font>(),<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span></span>)</span>) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_1_1_2"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E2:31_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T235" onmouseover="rs('xreal_1/T235')" onmouseout="rh()">XREAL_1:235</a></span>;<br/>
<a NAME="E3:31_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span>
 
;<br/>
<a NAME="E4:31_1_1_2"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E5:31_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1_2"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31_1_1_2"/>
<span class="p1">(<span class="default">(<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">(<font color="Maroon">F<sub>2</sub></font>(),<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span></span>)</span>) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">j</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">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T14" target="_self" onmouseover="rs('flexary1/T14')" onmouseout="rh()">Th14</a>, <a class="txt" href="#E1:31_1_1_2"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E3:31_1_1_2"><span class="lab"><font color="Green" title="E27">A8</font></span></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>
<span class="kw">hence </span><a NAME="E3:31_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E25">A4</font></span></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="E3:31_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
 for <font color="Olive" title="b1">j</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">j</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="#E1:31_1"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>);<br/></span>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>() or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c1">l</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E2:31_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E3:31_1_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T38" onmouseover="rs('xreal_1/T38')" onmouseout="rh()">XREAL_1:38</a></span>;<br/><a NAME="E4:31_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A11</font></span>: 
(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T16" target="_self" onmouseover="rs('flexary1/T16')" onmouseout="rh()">Th16</a></span>;<br/><a NAME="E5:31_1_2_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:31_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E7:31_1_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:31_1_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T38" onmouseover="rs('xreal_1/T38')" onmouseout="rh()">XREAL_1:38</a></span>;<br/><a NAME="E9:31_1_2_1"/><span class="kw">then </span>
( (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>) &amp; (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T16" target="_self" onmouseover="rs('flexary1/T16')" onmouseout="rh()">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E10:31_1_2_1"/>
(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E4:31_1_2_1"><span class="lab"><font color="Green" title="E27">A11</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_2"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c1">l</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>();<br/><a NAME="E2:31_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E3:31_1_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T38" onmouseover="rs('xreal_1/T38')" onmouseout="rh()">XREAL_1:38</a></span>;<br/><a NAME="E4:31_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A14</font></span>: 
(<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T16" target="_self" onmouseover="rs('flexary1/T16')" onmouseout="rh()">Th16</a></span>;<br/><a NAME="E5:31_1_2_2"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_2"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:31_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E7:31_1_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1_2_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:31_1_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T38" onmouseover="rs('xreal_1/T38')" onmouseout="rh()">XREAL_1:38</a></span>;<br/><a NAME="E9:31_1_2_2"/><span class="kw">then </span>
(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T16" target="_self" onmouseover="rs('flexary1/T16')" onmouseout="rh()">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E10:31_1_2_2"/>
(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E4:31_1_2_2"><span class="lab"><font color="Green" title="E27">A14</font></span></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></div><span class="kw">end;</span></div>

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