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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">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>  st  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 holds <br/> ex <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">w2</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b2">w1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b3">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></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"> (  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 implies  ex <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w2</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b2">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:77"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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">  ex <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w2</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b2">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">X9</font> = <font color="Maroon" title="c1">X</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">w9</font> = <font color="Maroon" title="c2">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c3">X9</font></span>)</span> ;<br/>
<a NAME="E4:77"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c4">w9</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"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></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="algstr_4.html#T25" target="_self" onmouseover="rs('algstr_4/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">n</font> =  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font>;<br/>
<a NAME="E5:77"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">fs</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E7:77"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E8:77"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/><font color="Maroon" title="c6">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><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="Olive" title="b1">p</font></span>)</span>,<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:77"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<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> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="algstr_4.html#D13" target="_self" onmouseover="rs('algstr_4/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E10:77"/>
<font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</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> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E11:77"/><span class="kw">then </span>
<font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <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> <font color="Maroon" title="c6">fs</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:77"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <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="c7">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E13:77"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
( <font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c6">fs</font></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="c8">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E15:77"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
( <font color="Maroon" title="c8">p</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> <font color="Maroon" title="c6">fs</font></span>)</span> &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c6">fs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:77"><span class="lab"><font color="Green" title="E45">A7</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="E16:77"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">fs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:77"><span class="lab"><font color="Green" title="E46">A8</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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c8">p</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E18:77"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">fs</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:77"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E19:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A11</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">fs</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E20:77"/><span class="kw">then </span>
<font color="Maroon" title="c6">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><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="c9">p</font></span>)</span>,<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</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="#E7:77"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E8:77"><span class="lab"><font color="Green" title="E43">A5</font></span></a></span>;<br/>
<a NAME="E21:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <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"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><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="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">p</font></span>)</span>,<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">p</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="#E13:77"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E15:77"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E16:77"><span class="lab"><font color="Green" title="E47">A9</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="E22:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A13</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <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"><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="c9">p</font></span>)</span>,<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <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="c9">p</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/>
<a NAME="E23:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A14</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</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> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">p</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <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">(<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">p</font></span>)</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/>
<a NAME="E24:77"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:77"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E7:77"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E25:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A15</font></span>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E26:77"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">p</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="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">m</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">p</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">w19</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">w29</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>,<font color="Maroon" title="c10">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E28:77"/>
<font color="Maroon" title="c9">p</font> <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="c9">p</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E29:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c9">p</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">p</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="#E23:77"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E30:77"/>
<font color="Maroon" title="c10">m</font> <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="c10">m</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E31:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>,<font color="Maroon" title="c10">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c10">m</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</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="#E23:77"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E32:77"/>
<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="c9">p</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">p</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E19:77"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="algstr_4.html#E39"><span class="lab"><font color="Green" title="E28">Lm1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">w19</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="txt" href="#E29:77"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>;<br/>
<a NAME="E34:77"/>
<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="c10">m</font>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E25:77"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="algstr_4.html#E39"><span class="lab"><font color="Green" title="E28">Lm1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">w29</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>,<font color="Maroon" title="c10">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="txt" href="#E31:77"><span class="lab"><font color="Green" title="E55">A17</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">w1</font> = <font color="Maroon" title="c13">w19</font>, <font color="Maroon" title="c16">w2</font> = <font color="Maroon" title="c14">w29</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E56">A18</font></span>:  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c15">w1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
<font color="Maroon" title="c9">p</font>

;<br/>
<span class="lab"><font color="Green" title="E57">A19</font></span>:  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c16">w2</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>,<font color="Maroon" title="c10">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
<font color="Maroon" title="c10">m</font>

;<br/>
<a NAME="E39:77"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">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"><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="c9">p</font></span>)</span>,<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:77"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E40:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A20</font></span>: 
<font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<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>;<br/>
<a NAME="E41:77"/><span class="lab"><font color="Green" title="E59">A21</font></span>: 
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:77"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">w1</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">  ex <font color="Olive" title="b1">w2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">w2</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c15">w1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c16">w2</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" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">w2</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c15">w1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c16">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> )</span><br/>

<a NAME="E42:77"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>) &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">w9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c2">w</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:77"><span class="lab"><font color="Green" title="E58">A20</font></span></a>, <a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">w1</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c15">w1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c16">w2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:77"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E38:77"><span class="lab"><font color="Green" title="E57">A19</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">w1</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c15">w1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c16">w2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:77"><span class="lab"><font color="Green" title="E59">A21</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">w1</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c16">w2</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">w1</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c15">w1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c16">w2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

<br/>.= 
<font color="Maroon" title="c15">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">w2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T31" target="_self" onmouseover="rs('algstr_4/T31')" onmouseout="rh()">Th31</a></span>
;<br/> <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="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c15">w1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c16">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> )</span><br/>

<a NAME="E44:77"/>
<font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:77"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E7:77"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E45:77"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E46:77"/>
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c15">w1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:77"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</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">  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c16">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span><br/>

<a NAME="E47:77"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:77"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E48:77"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font></span>)</span> <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="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E49:77"/><span class="kw">then </span>
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<span class="kw">hence </span><a NAME="E50:77"/>
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c16">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c2">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:77"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</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>
