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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">n</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:59"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN: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"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">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> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN: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> )</span><br/>

<span class="kw">assume </span><a NAME="E2:59"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN: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"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">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>
 ; <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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN: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></span><br/>

<a NAME="E3:59"/>
<font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:59"/>
<font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI: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="txt" href="#E1:59"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="nat_lat.html#D6" onmouseover="rs('nat_lat/D6')" onmouseout="rh()">NAT_LAT:def 6</a></span>;<br/>
<a NAME="E6:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI: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="#E4:59"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <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="lab"><font color="Green" title="E33">A5</font></span>: <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="c2">n</font> = 
<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="c2">n</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">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="ref" href="card_3.html#D3" onmouseover="rs('card_3/D3')" onmouseout="rh()">CARD_3:def 3</a></span>
<br/>.= 
<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"><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="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">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="#E6:59"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
;<br/>
<a NAME="E8:59"/>
<font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E32">A4</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="E9:59"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">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> <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="txt" href="#E7:59"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E10:59"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN: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="txt" href="#E2:59"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:59"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN: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>
 <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>; <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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