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<a NAME="E1:62_2_1"/>
<font color="Maroon" title="c2">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E2:62_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K10" title="ALGSTR_4:func.10">free_magma_seq</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:62_2_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
( <font color="Maroon" title="c2">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K10" title="ALGSTR_4:func.10">free_magma_seq</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:62_2_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K10" title="ALGSTR_4:func.10">free_magma_seq</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K10" title="ALGSTR_4:func.10">free_magma_seq</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_2_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:62_2_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K10" title="ALGSTR_4:func.10">free_magma_seq</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_2_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="card_3.html#D3" onmouseover="rs('card_3/D3')" onmouseout="rh()">CARD_3:def 3</a></span>;<br/>
<a NAME="E8:62_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T57" onmouseover="rs('relat_1/T57')" onmouseout="rh()">RELAT_1:57</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E10:62_2_1"/>
<font color="Maroon" title="c2">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K10" title="ALGSTR_4:func.10">free_magma_seq</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_2_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E4:62_2_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E7:62_2_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="card_3.html#D3" onmouseover="rs('card_3/D3')" onmouseout="rh()">CARD_3:def 3</a></span>;<br/>
<a NAME="E11:62_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:62_2_1"/>
 for <font color="Olive">b<sub>1</sub></font> being    <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">number</a>   st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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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