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<span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</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">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">x</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">n</font>)</span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> ) ) ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">x</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">n</font>)</span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> ) ) ) implies <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_2"/><span class="lab"><font color="Green" title="E1">A9</font></span>: 
 for <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">h</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">h</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">n</font>)</span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> ) )
 <span class="kw">and </span><br/><a NAME="E2:1_1_2"/><span class="lab"><font color="Green" title="E2">A10</font></span>: 
 for <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">h</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">h</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">n</font>)</span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</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"> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="1_1_2_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">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">n</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<font color="Olive" title="b1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<font color="Olive" title="b1">n</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">n</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><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><div><span class="lab"><font color="Green" title="E3">A11</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="1_1_2_1_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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:1_1_2_1_1"/><span class="lab"><font color="Green" title="E4">A12</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><a NAME="E2:1_1_2_1_1"/><span class="lab"><font color="Green" title="E5">A13</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c6">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c6">n</font>)
 ;<br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)

<br/>.= 
<font color="Maroon" title="c4">h</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c6">n</font>)</span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_2"><span class="lab"><font color="Green" title="E1">A9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_2"><span class="lab"><font color="Green" title="E2">A10</font></span></a>, <a class="txt" href="#E1:1_1_2_1_1"><span class="lab"><font color="Green" title="E4">A12</font></span></a>, <a class="txt" href="#E2:1_1_2_1_1"><span class="lab"><font color="Green" title="E5">A13</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><span class="kw">hence </span><a NAME="E4:1_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)

<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_2"><span class="lab"><font color="Green" title="E1">A9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_2"><span class="lab"><font color="Green" title="E2">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><a NAME="E3:1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A14</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 ;<br/><a NAME="E4:1_1_2_1"/>
 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="#E3:1_1_2_1"><span class="lab"><font color="Green" title="E4">A14</font></span></a>, <a class="txt" href="#E1:1_1_2_1"><span class="lab"><font color="Green" title="E3">A11</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E5:1_1_2_1"/>
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">h</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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</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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