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<b>let </b><font color="Maroon" title="c2">Y0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">Y0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</a> implies  not <font color="Maroon" title="c2">Y0</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/>

<b>assume </b><a NAME="E1:30_2_1"/><i><font color="Green" title="E11">A3</font></i>: 
not <font color="Maroon" title="c2">Y0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c2">Y0</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/>

<b>assume </b><a NAME="E2:30_2_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c2">Y0</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> implies <font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> )</span><br/><b>assume </b><a NAME="E1:30_2_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a></span><br/><b>then reconsider </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c3">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> ;<br/><b>reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="tex_2.html#E2:30_2_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="tex_2.html#E1:30_2_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/><a NAME="E4:30_2_1_1"/>
<font color="Maroon" title="c5">B</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 ;<br/><b>hence </b><a NAME="E5:30_2_1_1"/>
<font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_2.html#E1:30_2_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="tex_2.html#D3" target="_self" title="TEX_2:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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