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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>  st <font color="Olive" title="b1">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  holds <br/>( <span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">H</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <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">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  implies ( <span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  ) )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">P1</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">P2</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<span class="kw">assume </span><a NAME="E3:72"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c3">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  )</span><br/>

<a NAME="E4:72"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c5">P2</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">Y1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P2</font> or <font color="Maroon" title="c6">Y1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_1"/>
<font color="Maroon" title="c6">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P2</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="c6">Y1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:72_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">and </span><br/><a NAME="E4:72_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c6">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T58" target="_self" title="BORSUK_1:th.58">Th58</a></span>;<br/>
<a NAME="E5:72_1"/>
<font color="Maroon" title="c7">D</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E3:72_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:72_1"/>
<font color="Maroon" title="c6">Y1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="borsuk_1.html#T59" target="_self" title="BORSUK_1:th.59">Th59</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/>


</div><span class="kw">end;</span></div>
<a NAME="E5:72"/>
<font color="Maroon" title="c4">P1</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">X1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P1</font> or <font color="Maroon" title="c6">X1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_2"/>
<font color="Maroon" title="c6">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P1</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="c6">X1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:72_2"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">and </span><br/><a NAME="E4:72_2"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T57" target="_self" title="BORSUK_1:th.57">Th57</a></span>;<br/>
<a NAME="E5:72_2"/>
<font color="Maroon" title="c7">D</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E3:72_2"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:72_2"/>
<font color="Maroon" title="c6">X1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_2"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="borsuk_1.html#T59" target="_self" title="BORSUK_1:th.59">Th59</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:72"/>
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E46">A2</font></span></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/>


</div>
