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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="polnot_1.html#NM3" title="POLNOT_1:NM.3">Polish-language</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">p</font>, <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b3">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> 2  st <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> holds <br/> <a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Olive" title="b3">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</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">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> 2  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font> holds <br/> <a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Olive" title="b1">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> 2; <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="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font> implies  <a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:131"/><span class="lab"><font color="Green" title="E76">A1</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</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">  <a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">w</font> =  <a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Maroon" title="c4">r</font>);<br/>
<a NAME="E2:131"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:131"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E5:131"/><span class="lab"><font color="Green" title="E78">A3</font></span>: 
<span class="p1">(<span class="default"><a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D32" target="_self" onmouseover="rs('polnot_1/D32')" onmouseout="rh()">Def32</a></span>;<br/>
<span class="lab"><font color="Green" title="E79">A5</font></span>: <span class="p1">(<span class="default"><a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <font color="Maroon" title="c4">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E78">A3</font></span></a>, <a class="ref" href="polnot_1.html#T58" target="_self" onmouseover="rs('polnot_1/T58')" onmouseout="rh()">Th58</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E76">A1</font></span></a></span>
;<br/>
<a NAME="E7:131"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:131"/><span class="lab"><font color="Green" title="E80">A7</font></span>: 
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E10:131"/><span class="lab"><font color="Green" title="E81">A8</font></span>: 
<span class="p1">(<span class="default"><a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D32" target="_self" onmouseover="rs('polnot_1/D32')" onmouseout="rh()">Def32</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:131"/>
 <a href="polnot_1.html#K28" title="POLNOT_1:func.28">decomp</a> (<font color="Maroon" title="c1">S</font>,2,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="txt" href="#E6:131"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="txt" href="#E9:131"><span class="lab"><font color="Green" title="E80">A7</font></span></a>, <a class="txt" href="#E10:131"><span class="lab"><font color="Green" title="E81">A8</font></span></a>, <a class="ref" href="polnot_1.html#T62" target="_self" onmouseover="rs('polnot_1/T62')" onmouseout="rh()">Th62</a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</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/>


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