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<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">a2</font>, <font color="Maroon" title="c9">a3</font>, <font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c11">b2</font>, <font color="Maroon" title="c12">b3</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="c5">a</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>  &amp; <font color="Maroon" title="c6">b</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>  &amp; <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c8">a2</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>  &amp; <font color="Maroon" title="c9">a3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c10">b1</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>  &amp; <font color="Maroon" title="c11">b2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c12">b3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:187"/><span class="lab"><font color="Green" title="E130">A1</font></span>: 
( <font color="Maroon" title="c5">a</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>  &amp; <font color="Maroon" title="c6">b</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>  &amp; <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c8">a2</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>  &amp; <font color="Maroon" title="c9">a3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c10">b1</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>  &amp; <font color="Maroon" title="c11">b2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c12">b3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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">  <a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">f</font> =  <a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c15">x2</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="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> or  not <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> or  not <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</font> or <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:187"/><span class="lab"><font color="Green" title="E131">A2</font></span>: 
( <font color="Maroon" title="c14">x1</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="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> &amp; <font color="Maroon" title="c15">x2</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="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> )
 <span class="kw">and </span><br/><a NAME="E3:187"/><span class="lab"><font color="Green" title="E132">A3</font></span>: 
<span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</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="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c17">x2</font> = <font color="Maroon" title="c15">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:187"><span class="lab"><font color="Green" title="E131">A2</font></span></a>, <a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="187_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c16">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">x2</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:187_1"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">x2</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"> contradiction</span><br/><a NAME="E2:187_1"/><span class="kw">then </span>
( <font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">x2</font> or <font color="Maroon" title="c17">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:187_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:187"><span class="lab"><font color="Green" title="E130">A1</font></span></a>, <a class="ref" href="number08.html#T91" target="_self" onmouseover="rs('number08/T91')" onmouseout="rh()">Th91</a></span>;<br/><span class="kw">hence </span><a NAME="E4:187_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:187"><span class="lab"><font color="Green" title="E132">A3</font></span></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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:187"/>
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
