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<span class="kw">let </span><font color="Maroon" title="c3">Z1</font>, <font color="Maroon" title="c4">Z2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="fintopo6.html#M1" title="FINTOPO6:mode.1">SubSpace</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">Z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> implies <font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1_2"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E2:15_1_2"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">Z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">R1</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">Z1</font>, <font color="Maroon" title="c6">R2</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Z2</font> as   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a></span>;<br/>
<a NAME="E4:15_1_2"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> holds <br/> <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c5">R1</font>,<font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c6">R2</font>,<font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> implies  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c5">R1</font>,<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c6">R2</font>,<font color="Maroon" title="c7">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_2_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c5">R1</font>,<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c6">R2</font>,<font color="Maroon" title="c7">z</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">z1</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Z1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">z2</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">Z2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E1:15_1_2_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a></span>;<br/>
<span class="kw">thus </span> <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c5">R1</font>,<font color="Maroon" title="c7">z</font> = 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c8">z1</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c8">z1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Z1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo6.html#D2" target="_self" title="FINTOPO6:def.2">Def2</a></span>
<br/>.= 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c9">z2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="fintopo6.html#D2" target="_self" title="FINTOPO6:def.2">Def2</a></span>
<br/>.= 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c6">R2</font>,<font color="Maroon" title="c7">z</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:15_1_2"/>
<font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="relset_1.html#T54" title="RELSET_1:th.54">RELSET_1:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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