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<span class="kw">set </span><font color="Maroon" title="c1">ZS</font> =  <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</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">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">t</font> implies  not <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E19">AS</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">t</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">  not <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:35"/><span class="lab"><font color="Green" title="E20">P1</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</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"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:35"/><span class="lab"><font color="Green" title="E21">P2</font></span>: 
( <font color="Maroon" title="c5">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>  &amp; <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rat_1.html#T1" onmouseover="rs('rat_1/T1')" onmouseout="rh()">RAT_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E6:35"/><span class="lab"><font color="Green" title="E22">P3</font></span>: 
( <font color="Maroon" title="c7">q</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="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rat_1.html#T1" onmouseover="rs('rat_1/T1')" onmouseout="rh()">RAT_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" onmouseover="rs('int_1/D2')" onmouseout="rh()">INT_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" onmouseover="rs('int_1/D2')" onmouseout="rh()">INT_1:def 2</a></span>;<br/>
<a NAME="E9:35"/><span class="lab"><font color="Green" title="E23">P4</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">AS</font></span></a>, <a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E20">P1</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E22">P3</font></span></a>, <a class="ref" href="zmodul02.html#T62" onmouseover="rs('zmodul02/T62')" onmouseout="rh()">ZMODUL02:62</a></span>;<br/>
<font color="Maroon" title="c8">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T15" target="_self" onmouseover="rs('zmodul07/T15')" onmouseout="rh()">LMTFRat2</a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E21">P2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E21">P2</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" onmouseover="rs('xcmplx_1/T90')" onmouseout="rh()">XCMPLX_1:90</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E22">P3</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" onmouseover="rs('xcmplx_1/T90')" onmouseout="rh()">XCMPLX_1:90</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T15" target="_self" onmouseover="rs('zmodul07/T15')" onmouseout="rh()">LMTFRat2</a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E22">P3</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:35"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">AS</font></span></a>, <a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E20">P1</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E21">P2</font></span></a>, <a class="txt" href="#E9:35"><span class="lab"><font color="Green" title="E23">P4</font></span></a>, <a class="ref" href="zmodul02.html#T62" onmouseover="rs('zmodul02/T62')" onmouseout="rh()">ZMODUL02:62</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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