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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="mfold_0.html#V1" title="MFOLD_0:attr.1">locally_euclidean</a>   <a href="mfold_0.html#NV4" title="MFOLD_0:NV.4">with_boundary</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  ex <font color="Olive" title="b3">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">p</font> st <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">U</font>, <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  holds <br/> for <font color="Olive" title="b3">pF</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">pF</font> holds <br/> ex <font color="Olive" title="b4">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b3">pF</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b4">U</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</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">  for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  ex <font color="Olive" title="b2">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">p</font> st <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">U</font>, <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  holds <br/> for <font color="Olive" title="b2">pF</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>  st <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">pF</font> holds <br/> ex <font color="Olive" title="b3">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b2">pF</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">U</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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"> (  ex <font color="Olive" title="b1">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">p</font> st <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font>, <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  implies  for <font color="Olive" title="b1">pF</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>  st <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pF</font> holds <br/> ex <font color="Olive" title="b2">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">pF</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">U</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 ex <font color="Olive" title="b1">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">p</font> st <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font>, <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</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">  for <font color="Olive" title="b1">pF</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>  st <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pF</font> holds <br/> ex <font color="Olive" title="b2">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">pF</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">U</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">TR</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">p</font><span class="kw"> such that </span><br/><a NAME="E3:30"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">W</font>, <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E12">A1</font></span></a></span>;<br/>
<a NAME="E4:30"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" onmouseover="rs('connsp_2/D1')" onmouseout="rh()">CONNSP_2:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">TR1</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">CL</font> =  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">S</font> =  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">F</font> =  <a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">MF</font> = <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">pF</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</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"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">pF</font> implies  ex <font color="Olive" title="b1">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c12">pF</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E5:30"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">pF</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">  ex <font color="Olive" title="b1">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c12">pF</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> </span><br/>

<a NAME="E6:30"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c14">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c15">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span>,1)</span>)</span><span class="kw"> such that </span><br/><a NAME="E8:30"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c15">h</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E9:30"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span>,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="mfold_0.html#T5" target="_self" onmouseover="rs('mfold_0/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E10:30"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" onmouseover="rs('connsp_2/D1')" onmouseout="rh()">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E11:30"/>
<font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font>, <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="t_0topsp.html#D1" onmouseover="rs('t_0topsp/D1')" onmouseout="rh()">T_0TOPSP:def 1</a></span>;<br/>
<a NAME="E12:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a>, <a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="brouwer3.html#T14" onmouseover="rs('brouwer3/T14')" onmouseout="rh()">BROUWER3:14</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">h</font> = <font color="Maroon" title="c15">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> ;<br/>
<a NAME="E14:30"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T16" onmouseover="rs('tops_1/T16')" onmouseout="rh()">TOPS_1:16</a></span>;<br/>
<a NAME="E15:30"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">IU</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T16" onmouseover="rs('tops_1/T16')" onmouseout="rh()">TOPS_1:16</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">MU</font> = <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font>;<br/>
<a NAME="E17:30"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c12">pF</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="connsp_2.html#D1" onmouseover="rs('connsp_2/D1')" onmouseout="rh()">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E18:30"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">IU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">FIU</font> = <span class="p1">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font></span>)</span> 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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> ;<br/>
<a NAME="E20:30"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E21:30"/>
<font color="Maroon" title="c17">IU</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="ref" href="tsep_1.html#T9" onmouseover="rs('tsep_1/T9')" onmouseout="rh()">TSEP_1:9</a></span>;<br/>
<a NAME="E22:30"/><span class="kw">then </span>
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</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="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="topgrp_1.html#T25" onmouseover="rs('topgrp_1/T25')" onmouseout="rh()">TOPGRP_1:25</a></span>;<br/>
<a NAME="E23:30"/><span class="kw">then </span>
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E25:30"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c20">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E26:30"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D4" onmouseover="rs('pre_topc/D4')" onmouseout="rh()">PRE_TOPC:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">W</font> = <font color="Maroon" title="c20">W</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:30"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E28:30"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a>, <a class="txt" href="#E26:30"><span class="lab"><font color="Green" title="E25">A14</font></span></a></span>;<br/>
<a NAME="E29:30"/><span class="lab"><font color="Green" title="E27">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/>
<a NAME="E30:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A17</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E17:30"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">hp</font> = <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30"><span class="lab"><font color="Green" title="E26">A15</font></span></a></span>;<br/>
<a NAME="E32:30"/><span class="lab"><font color="Green" title="E29">A18</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c21">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T23" onmouseover="rs('tops_1/T23')" onmouseout="rh()">TOPS_1:23</a></span>;<br/>
<a NAME="E33:30"/><span class="lab"><font color="Green" title="E30">A19</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">hp</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="topreal9.html#T9" onmouseover="rs('topreal9/T9')" onmouseout="rh()">TOPREAL9:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">HP</font> = <font color="Maroon" title="c22">hp</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T67" onmouseover="rs('euclid/T67')" onmouseout="rh()">EUCLID:67</a></span>;<br/>
<a NAME="E35:30"/>
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:30"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="txt" href="#E26:30"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E37:30"/><span class="lab"><font color="Green" title="E31">A20</font></span>: 
<font color="Maroon" title="c24">s</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> 
 <span class="kw">and </span><br/><a NAME="E38:30"/><span class="lab"><font color="Green" title="E32">A21</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c23">HP</font>,<font color="Maroon" title="c24">s</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:30"><span class="lab"><font color="Green" title="E29">A18</font></span></a>, <a class="ref" href="goboard6.html#T5" onmouseover="rs('goboard6/T5')" onmouseout="rh()">GOBOARD6:5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">s2</font> = <font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">m</font> =  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c27">V0</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">hV0</font> = <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>;<br/>
<a NAME="E39:30"/><span class="lab"><font color="Green" title="E33">A22</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/>
<a NAME="E40:30"/>
<font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:30"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</a></span>;<br/>
<a NAME="E41:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A23</font></span>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<font color="Maroon" title="c24">s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:30"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="ref" href="jordan.html#T18" onmouseover="rs('jordan/T18')" onmouseout="rh()">JORDAN:18</a></span>;<br/>
<a NAME="E42:30"/><span class="lab"><font color="Green" title="E35">A24</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c23">HP</font>,<font color="Maroon" title="c24">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<font color="Maroon" title="c24">s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T13" onmouseover="rs('topreal9/T13')" onmouseout="rh()">TOPREAL9:13</a></span>;<br/>
<a NAME="E43:30"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">x</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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> or <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</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"> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span><br/>

<a NAME="E2:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A27</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E3:30_1"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">q</font> = <font color="Maroon" title="c29">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:30"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E23">A12</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">hq</font> = <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E38">A27</font></span></a></span>;<br/>
<a NAME="E6:30_1"/>
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A29</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<font color="Maroon" title="c24">s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:30"><span class="lab"><font color="Green" title="E34">A23</font></span></a></span>;<br/>
<a NAME="E8:30_1"/><span class="lab"><font color="Green" title="E41">A30</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:30_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">hq</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T9" onmouseover="rs('topreal9/T9')" onmouseout="rh()">TOPREAL9:9</a></span>;<br/>
<a NAME="E10:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c31">hq</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
;<br/>
<a NAME="E11:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E38:30"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="txt" href="#E42:30"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E13:30_1"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">h</font>
 <span class="kw">and </span><br/><a NAME="E14:30_1"/><span class="lab"><font color="Green" title="E43">A32</font></span>: 
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">IU</font>
 <span class="kw">and </span><br/><a NAME="E15:30_1"/><span class="lab"><font color="Green" title="E44">A33</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E16:30_1"/>
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E13:30_1"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="txt" href="#E15:30_1"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E17:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c30">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="ref" href="connsp_2.html#D1" onmouseover="rs('connsp_2/D1')" onmouseout="rh()">CONNSP_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:30_1"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="mfold_0.html#T5" target="_self" onmouseover="rs('mfold_0/T5')" onmouseout="rh()">Th5</a>, <a class="txt" href="#E8:30_1"><span class="lab"><font color="Green" title="E41">A30</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">reconsider </span><font color="Maroon" title="c29">FIU1</font> = <font color="Maroon" title="c19">FIU</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a>, <a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<a NAME="E45:30"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E46:30"/><span class="kw">then </span>
<font color="Maroon" title="c29">FIU1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="pre_topc.html#D4" onmouseover="rs('pre_topc/D4')" onmouseout="rh()">PRE_TOPC:def 4</a></span>;<br/>
<a NAME="E47:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A34</font></span>: 
<font color="Maroon" title="c29">FIU1</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="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E48:30"/><span class="lab"><font color="Green" title="E38">A35</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c29">FIU1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a>, <a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">Mfiu</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">FIU</font>;<br/>
<a NAME="E49:30"/><span class="lab"><font color="Green" title="E39">A36</font></span>: 
<span class="p1">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E50:30"/><span class="lab"><font color="Green" title="E40">A37</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">hFIU</font> = <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c19">FIU</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E52:30"/><span class="lab"><font color="Green" title="E41">A38</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">hFIU</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">hFIU</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E53:30"/><span class="lab"><font color="Green" title="E42">A39</font></span>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c19">FIU</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">hFIU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:30"><span class="lab"><font color="Green" title="E40">A37</font></span></a>, <a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">hfiu</font> = <font color="Maroon" title="c16">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c19">FIU</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">FIU</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">hFIU</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan24.html#T12" onmouseover="rs('jordan24/T12')" onmouseout="rh()">JORDAN24:12</a></span>;<br/>
<a NAME="E55:30"/><span class="lab"><font color="Green" title="E43">A40</font></span>: 
<font color="Maroon" title="c32">hfiu</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="metrizts.html#T2" onmouseover="rs('metrizts/T2')" onmouseout="rh()">METRIZTS:2</a>, <a class="txt" href="#E53:30"><span class="lab"><font color="Green" title="E42">A39</font></span></a></span>;<br/>
<a NAME="E56:30"/><span class="lab"><font color="Green" title="E44">A41</font></span>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T19" onmouseover="rs('topreal9/T19')" onmouseout="rh()">TOPREAL9:19</a></span>;<br/>
<a NAME="E57:30"/><span class="lab"><font color="Green" title="E45">A42</font></span>: 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T17" onmouseover="rs('topreal9/T17')" onmouseout="rh()">TOPREAL9:17</a></span>;<br/>
<a NAME="E58:30"/><span class="lab"><font color="Green" title="E46">A43</font></span>: 
<font color="Maroon" title="c17">IU</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T82" onmouseover="rs('funct_1/T82')" onmouseout="rh()">FUNCT_1:82</a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="funct_1.html#T76" onmouseover="rs('funct_1/T76')" onmouseout="rh()">FUNCT_1:76</a>, <a class="txt" href="#E29:30"><span class="lab"><font color="Green" title="E27">A16</font></span></a></span>;<br/>
<a NAME="E59:30"/>
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E60:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A44</font></span>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:30"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="txt" href="#E41:30"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="txt" href="#E42:30"><span class="lab"><font color="Green" title="E35">A24</font></span></a></span>;<br/>
<a NAME="E61:30"/><span class="lab"><font color="Green" title="E48">A45</font></span>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">hFIU</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">x</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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> or <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">hFIU</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E49">A46</font></span>: 
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</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"> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">hFIU</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">q</font> = <font color="Maroon" title="c33">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ;<br/>
<a NAME="E3:30_2"/>
<font color="Maroon" title="c34">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:30_2"/><span class="kw">then </span>
<font color="Maroon" title="c34">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:30"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="txt" href="#E28:30"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E57:30"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:30_2"/><span class="lab"><font color="Green" title="E50">A47</font></span>: 
<font color="Maroon" title="c35">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">h</font>
 <span class="kw">and </span><br/><a NAME="E7:30_2"/><span class="lab"><font color="Green" title="E51">A48</font></span>: 
<font color="Maroon" title="c35">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">IU</font>
 <span class="kw">and </span><br/><a NAME="E8:30_2"/><span class="lab"><font color="Green" title="E52">A49</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c36">w</font> = <font color="Maroon" title="c35">w</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E50">A47</font></span></a></span>;<br/>
<a NAME="E10:30_2"/>
<font color="Maroon" title="c36">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E50">A47</font></span></a>, <a class="txt" href="#E8:30_2"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E11:30_2"/><span class="kw">then </span>
<font color="Maroon" title="c36">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">FIU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:30"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="txt" href="#E7:30_2"><span class="lab"><font color="Green" title="E51">A48</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:30_2"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">hFIU</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E50">A47</font></span></a>, <a class="txt" href="#E8:30_2"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</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>
<a NAME="E62:30"/><span class="lab"><font color="Green" title="E49">A50</font></span>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E63:30"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:30"><span class="lab"><font color="Green" title="E45">A42</font></span></a></span>;<br/>
<a NAME="E64:30"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:30"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="txt" href="#E26:30"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a>, <a class="txt" href="#E50:30"><span class="lab"><font color="Green" title="E40">A37</font></span></a></span>;<br/>
<a NAME="E65:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A51</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">IU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:30"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="ref" href="relat_1.html#T143" onmouseover="rs('relat_1/T143')" onmouseout="rh()">RELAT_1:143</a></span>;<br/>
<a NAME="E66:30"/><span class="lab"><font color="Green" title="E51">A52</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/>
<a NAME="E67:30"/><span class="kw">then </span>
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T77" onmouseover="rs('funct_1/T77')" onmouseout="rh()">FUNCT_1:77</a>, <a class="txt" href="#E57:30"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E62:30"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="txt" href="#E50:30"><span class="lab"><font color="Green" title="E40">A37</font></span></a></span>;<br/>
<a NAME="E68:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A53</font></span>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p5">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p0">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p0">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a>, <a class="txt" href="#E57:30"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E62:30"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E69:30"/><span class="lab"><font color="Green" title="E53">A54</font></span>: 
 <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T18" onmouseover="rs('topreal9/T18')" onmouseout="rh()">TOPREAL9:18</a></span>;<br/>
<a NAME="E70:30"/><span class="lab"><font color="Green" title="E54">A55</font></span>: 
<font color="Maroon" title="c31">hFIU</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">x</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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">hFIU</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> or <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_3"/><span class="lab"><font color="Green" title="E55">A56</font></span>: 
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">hFIU</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</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"> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">q</font> = <font color="Maroon" title="c33">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ;<br/>
<a NAME="E3:30_3"/><span class="lab"><font color="Green" title="E56">A57</font></span>: 
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">hFIU</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:30_3"/><span class="lab"><font color="Green" title="E57">A58</font></span>: 
<font color="Maroon" title="c34">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c35">Q</font> = <font color="Maroon" title="c34">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T67" onmouseover="rs('euclid/T67')" onmouseout="rh()">EUCLID:67</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c36">WB</font> = <font color="Maroon" title="c21">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>;<br/>
<a NAME="E2:30_3_1"/><span class="lab"><font color="Green" title="E58">A59</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T23" onmouseover="rs('tops_1/T23')" onmouseout="rh()">TOPS_1:23</a></span>;<br/>
<a NAME="E3:30_3_1"/>
<font color="Maroon" title="c31">hFIU</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a>, <a class="ref" href="relat_1.html#T123" onmouseover="rs('relat_1/T123')" onmouseout="rh()">RELAT_1:123</a></span>;<br/>
<a NAME="E4:30_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A60</font></span>: 
<font color="Maroon" title="c34">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_3"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="txt" href="#E26:30"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:30_3_1"/>
 not <font color="Maroon" title="c34">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 ; <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="E6:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c34">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_3"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="txt" href="#E50:30"><span class="lab"><font color="Green" title="E40">A37</font></span></a>, <a class="txt" href="#E69:30"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E7:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c34">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_3_1"><span class="lab"><font color="Green" title="E59">A60</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">w</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:30_3_1"/><span class="lab"><font color="Green" title="E60">A61</font></span>: 
<font color="Maroon" title="c37">w</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> 
 <span class="kw">and </span><br/><a NAME="E10:30_3_1"/><span class="lab"><font color="Green" title="E61">A62</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c35">Q</font>,<font color="Maroon" title="c37">w</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_3_1"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="ref" href="goboard6.html#T5" onmouseover="rs('goboard6/T5')" onmouseout="rh()">GOBOARD6:5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c38">B</font> =  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>);<br/>
<a NAME="E11:30_3_1"/><span class="lab"><font color="Green" title="E62">A63</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c35">Q</font>,<font color="Maroon" title="c37">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T13" onmouseover="rs('topreal9/T13')" onmouseout="rh()">TOPREAL9:13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c39">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E13:30_3_1"/><span class="lab"><font color="Green" title="E63">A64</font></span>: 
<font color="Maroon" title="c39">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">h</font>
 <span class="kw">and </span><br/><a NAME="E14:30_3_1"/><span class="lab"><font color="Green" title="E64">A65</font></span>: 
<font color="Maroon" title="c39">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">FIU</font>
 <span class="kw">and </span><br/><a NAME="E15:30_3_1"/><span class="lab"><font color="Green" title="E65">A66</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a>, <a class="txt" href="#E3:30_3"><span class="lab"><font color="Green" title="E56">A57</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c40">u</font> = <font color="Maroon" title="c39">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_3_1"><span class="lab"><font color="Green" title="E64">A65</font></span></a></span>;<br/>
<a NAME="E17:30_3_1"/><span class="lab"><font color="Green" title="E66">A67</font></span>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:30"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="ref" href="xboole_1.html#T11" onmouseover="rs('xboole_1/T11')" onmouseout="rh()">XBOOLE_1:11</a></span>;<br/>
<a NAME="E18:30_3_1"/>
<font color="Maroon" title="c21">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E19:30_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A68</font></span>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_3_1"><span class="lab"><font color="Green" title="E61">A62</font></span></a>, <a class="txt" href="#E11:30_3_1"><span class="lab"><font color="Green" title="E62">A63</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c41">BB</font> =  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_3_1"><span class="lab"><font color="Green" title="E66">A67</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="txt" href="#E50:30"><span class="lab"><font color="Green" title="E40">A37</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c42">hBB</font> = <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c41">BB</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E22:30_3_1"/><span class="lab"><font color="Green" title="E68">A69</font></span>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E23:30_3_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c34">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T28" onmouseover="rs('toprns_1/T28')" onmouseout="rh()">TOPRNS_1:28</a></span>;<br/>
<a NAME="E24:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c34">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c41">BB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_3_1"><span class="lab"><font color="Green" title="E60">A61</font></span></a></span>;<br/>
<a NAME="E25:30_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A70</font></span>: 
<font color="Maroon" title="c40">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c42">hBB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_3_1"><span class="lab"><font color="Green" title="E63">A64</font></span></a>, <a class="txt" href="#E15:30_3_1"><span class="lab"><font color="Green" title="E65">A66</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E26:30_3_1"/>
<font color="Maroon" title="c41">BB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">BB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_3_1"><span class="lab"><font color="Green" title="E66">A67</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="txt" href="#E19:30_3_1"><span class="lab"><font color="Green" title="E67">A68</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<a NAME="E27:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">BB</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:30_3_1"><span class="lab"><font color="Green" title="E68">A69</font></span></a>, <a class="txt" href="#E50:30"><span class="lab"><font color="Green" title="E40">A37</font></span></a>, <a class="ref" href="pre_topc.html#D4" onmouseover="rs('pre_topc/D4')" onmouseout="rh()">PRE_TOPC:def 4</a></span>;<br/>
<a NAME="E28:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">BB</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="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E29:30_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A72</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c41">BB</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="ref" href="topgrp_1.html#T26" onmouseover="rs('topgrp_1/T26')" onmouseout="rh()">TOPGRP_1:26</a>, <a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a></span>;<br/>
<a NAME="E30:30_3_1"/>
<font color="Maroon" title="c21">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E31:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">BB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_3_1"><span class="lab"><font color="Green" title="E61">A62</font></span></a>, <a class="txt" href="#E11:30_3_1"><span class="lab"><font color="Green" title="E62">A63</font></span></a></span>;<br/>
<a NAME="E32:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">BB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">IU</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a>, <a class="txt" href="#E26:30"><span class="lab"><font color="Green" title="E25">A14</font></span></a></span>;<br/>
<a NAME="E33:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c41">BB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T143" onmouseover="rs('relat_1/T143')" onmouseout="rh()">RELAT_1:143</a>, <a class="txt" href="#E58:30"><span class="lab"><font color="Green" title="E46">A43</font></span></a></span>;<br/>
<a NAME="E34:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">hBB</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="#E14:30"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E29:30_3_1"><span class="lab"><font color="Green" title="E70">A72</font></span></a>, <a class="ref" href="tsep_1.html#T9" onmouseover="rs('tsep_1/T9')" onmouseout="rh()">TSEP_1:9</a></span>;<br/>
<a NAME="E35:30_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A73</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c42">hBB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">hBB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T23" onmouseover="rs('tops_1/T23')" onmouseout="rh()">TOPS_1:23</a></span>;<br/>
<a NAME="E36:30_3_1"/><span class="lab"><font color="Green" title="E72">A74</font></span>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">BB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:30"><span class="lab"><font color="Green" title="E40">A37</font></span></a>, <a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c43">hBB</font> = <font color="Maroon" title="c42">hBB</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c40">u</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:30_3_1"><span class="lab"><font color="Green" title="E71">A73</font></span></a>, <a class="txt" href="#E25:30_3_1"><span class="lab"><font color="Green" title="E69">A70</font></span></a>, <a class="ref" href="connsp_2.html#D1" onmouseover="rs('connsp_2/D1')" onmouseout="rh()">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E38:30_3_1"/><span class="lab"><font color="Green" title="E73">A75</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c43">hBB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">BB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T77" onmouseover="rs('funct_1/T77')" onmouseout="rh()">FUNCT_1:77</a>, <a class="txt" href="#E66:30"><span class="lab"><font color="Green" title="E51">A52</font></span></a></span>;<br/>
<a NAME="E39:30_3_1"/><span class="lab"><font color="Green" title="E74">A76</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c41">BB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c43">hBB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c44">hB</font> = <font color="Maroon" title="c16">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c43">hBB</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c43">hBB</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>)</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan24.html#T12" onmouseover="rs('jordan24/T12')" onmouseout="rh()">JORDAN24:12</a>, <a class="txt" href="#E36:30_3_1"><span class="lab"><font color="Green" title="E72">A74</font></span></a>, <a class="txt" href="#E38:30_3_1"><span class="lab"><font color="Green" title="E73">A75</font></span></a></span>;<br/>
<a NAME="E41:30_3_1"/>
<font color="Maroon" title="c44">hB</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E36:30_3_1"><span class="lab"><font color="Green" title="E72">A74</font></span></a>, <a class="txt" href="#E38:30_3_1"><span class="lab"><font color="Green" title="E73">A75</font></span></a>, <a class="txt" href="#E39:30_3_1"><span class="lab"><font color="Green" title="E74">A76</font></span></a>, <a class="ref" href="metrizts.html#T2" onmouseover="rs('metrizts/T2')" onmouseout="rh()">METRIZTS:2</a></span>;<br/>
<a NAME="E42:30_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A77</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c43">hBB</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>) <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#D1" onmouseover="rs('t_0topsp/D1')" onmouseout="rh()">T_0TOPSP:def 1</a></span>;<br/>
<a NAME="E43:30_3_1"/>
 <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<font color="Maroon" title="c34">q</font>,<font color="Maroon" title="c37">w</font>), <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_3_1"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="ref" href="mfold_0.html#T3" target="_self" onmouseover="rs('mfold_0/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E44:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c43">hBB</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:30_3_1"><span class="lab"><font color="Green" title="E75">A77</font></span></a>, <a class="txt" href="#E9:30_3_1"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="ref" href="borsuk_3.html#T3" onmouseover="rs('borsuk_3/T3')" onmouseout="rh()">BORSUK_3:3</a></span>;<br/>
<a NAME="E45:30_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A78</font></span>: 
<font color="Maroon" title="c40">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="mfold_0.html#K2" title="MFOLD_0:func.2">Int</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mfold_0.html#D4" target="_self" onmouseover="rs('mfold_0/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E46:30_3_1"/>
<font color="Maroon" title="c40">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_3_1"><span class="lab"><font color="Green" title="E64">A65</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E47:30_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="mfold_0.html#K2" title="MFOLD_0:func.2">Int</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E48:30_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a>, <a class="txt" href="#E45:30_3_1"><span class="lab"><font color="Green" title="E76">A78</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>
<a NAME="E5:30_3"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:30_3"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_3"><span class="lab"><font color="Green" title="E57">A58</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>

;<br/>
<a NAME="E72:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A79</font></span>: 
<font color="Maroon" title="c31">hFIU</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:30"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a>, <a class="txt" href="#E61:30"><span class="lab"><font color="Green" title="E48">A45</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">v0</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">hFIU</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:30"><span class="lab"><font color="Green" title="E41">A38</font></span></a>, <a class="txt" href="#E61:30"><span class="lab"><font color="Green" title="E48">A45</font></span></a></span>;<br/>
<a NAME="E74:30"/>
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E75:30"/><span class="kw">then </span>
<font color="Maroon" title="c33">v0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">hFIU</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:30"><span class="lab"><font color="Green" title="E55">A79</font></span></a>, <a class="ref" href="pre_topc.html#D4" onmouseover="rs('pre_topc/D4')" onmouseout="rh()">PRE_TOPC:def 4</a>, <a class="txt" href="#E52:30"><span class="lab"><font color="Green" title="E41">A38</font></span></a></span>;<br/>
<a NAME="E76:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A80</font></span>: 
<font color="Maroon" title="c33">v0</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="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E77:30"/><span class="lab"><font color="Green" title="E57">A81</font></span>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T18" onmouseover="rs('topreal9/T18')" onmouseout="rh()">TOPREAL9:18</a></span>;<br/>
<a NAME="E78:30"/><span class="lab"><font color="Green" title="E58">A83</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">hp</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">hp</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T27" onmouseover="rs('toprns_1/T27')" onmouseout="rh()">TOPRNS_1:27</a></span>;<br/>
<a NAME="E79:30"/><span class="lab"><font color="Green" title="E59">A84</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:30"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T21" onmouseover="rs('xxreal_0/T21')" onmouseout="rh()">XXREAL_0:21</a></span>;<br/>
<a NAME="E80:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A85</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">hp</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:30"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="txt" href="#E78:30"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/>
<a NAME="E81:30"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">hp</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">hp</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T28" onmouseover="rs('toprns_1/T28')" onmouseout="rh()">TOPRNS_1:28</a></span>;<br/>
<a NAME="E82:30"/><span class="kw">then </span>
<font color="Maroon" title="c22">hp</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:30"><span class="lab"><font color="Green" title="E59">A84</font></span></a></span>;<br/>
<a NAME="E83:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A86</font></span>: 
<font color="Maroon" title="c22">hp</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E84:30"/><span class="lab"><font color="Green" title="E62">A87</font></span>: 
<font color="Maroon" title="c12">pF</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="connsp_2.html#D1" onmouseover="rs('connsp_2/D1')" onmouseout="rh()">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E85:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A88</font></span>: 
<font color="Maroon" title="c12">pF</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:30"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E83:30"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E86:30"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/>
<a NAME="E87:30"/><span class="kw">then </span>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">hp</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:30"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="txt" href="#E78:30"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)</span>)</span><span class="kw"> such that </span><br/><a NAME="E89:30"/><span class="lab"><font color="Green" title="E64">A89</font></span>: 
<font color="Maroon" title="c34">g</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E90:30"/><span class="lab"><font color="Green" title="E65">A90</font></span>: 
<font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:30"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="txt" href="#E78:30"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E80:30"><span class="lab"><font color="Green" title="E60">A85</font></span></a>, <a class="ref" href="brouwer3.html#T19" onmouseover="rs('brouwer3/T19')" onmouseout="rh()">BROUWER3:19</a></span>;<br/>
<a NAME="E91:30"/><span class="lab"><font color="Green" title="E66">A91</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:30"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="ref" href="funct_1.html#T62" onmouseover="rs('funct_1/T62')" onmouseout="rh()">FUNCT_1:62</a></span>;<br/>
<a NAME="E92:30"/><span class="lab"><font color="Green" title="E67">A92</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">FIU</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">FIU</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">U0</font> = <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> 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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">FIU</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:30"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E43:30"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a>, <a class="txt" href="#E29:30"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="txt" href="#E84:30"><span class="lab"><font color="Green" title="E62">A87</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E83:30"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c36">U0</font> = <font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">FIU</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:30"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E43:30"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a>, <a class="txt" href="#E92:30"><span class="lab"><font color="Green" title="E67">A92</font></span></a></span>;<br/>
<a NAME="E95:30"/><span class="lab"><font color="Green" title="E68">A93</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c29">FIU1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">FIU</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c37">u0</font> = <font color="Maroon" title="c36">U0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c29">FIU1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E92:30"><span class="lab"><font color="Green" title="E67">A92</font></span></a></span>;<br/>
<a NAME="E97:30"/>
<font color="Maroon" title="c32">hfiu</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c33">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">FIU</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c36">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" onmouseover="rs('funct_1/T70')" onmouseout="rh()">FUNCT_1:70</a></span>;<br/>
<a NAME="E98:30"/><span class="kw">then </span>
<font color="Maroon" title="c32">hfiu</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c33">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:30"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E43:30"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<a NAME="E99:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A94</font></span>: 
<font color="Maroon" title="c36">U0</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="#E76:30"><span class="lab"><font color="Green" title="E56">A80</font></span></a>, <a class="txt" href="#E55:30"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="ref" href="topgrp_1.html#T26" onmouseover="rs('topgrp_1/T26')" onmouseout="rh()">TOPGRP_1:26</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c38">FV0</font> = <font color="Maroon" title="c37">u0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="txt" href="#E92:30"><span class="lab"><font color="Green" title="E67">A92</font></span></a></span>;<br/>
<a NAME="E101:30"/><span class="lab"><font color="Green" title="E70">A95</font></span>: 
<span class="p1">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a>, <a class="ref" href="tops_1.html#T16" onmouseover="rs('tops_1/T16')" onmouseout="rh()">TOPS_1:16</a></span>;<br/>
<a NAME="E102:30"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">FIU</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c13">U</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:30"><span class="lab"><font color="Green" title="E39">A36</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<a NAME="E103:30"/><span class="kw">then </span>
<font color="Maroon" title="c38">FV0</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="#E47:30"><span class="lab"><font color="Green" title="E37">A34</font></span></a>, <a class="txt" href="#E48:30"><span class="lab"><font color="Green" title="E38">A35</font></span></a>, <a class="txt" href="#E99:30"><span class="lab"><font color="Green" title="E69">A94</font></span></a>, <a class="ref" href="tsep_1.html#T9" onmouseover="rs('tsep_1/T9')" onmouseout="rh()">TSEP_1:9</a>, <a class="txt" href="#E95:30"><span class="lab"><font color="Green" title="E68">A93</font></span></a></span>;<br/>
<a NAME="E104:30"/><span class="kw">then </span>
<font color="Maroon" title="c12">pF</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c38">FV0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:30"><span class="lab"><font color="Green" title="E63">A88</font></span></a>, <a class="ref" href="tops_1.html#T22" onmouseover="rs('tops_1/T22')" onmouseout="rh()">TOPS_1:22</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c39">FV0</font> = <font color="Maroon" title="c38">FV0</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c12">pF</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" onmouseover="rs('connsp_2/D1')" onmouseout="rh()">CONNSP_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c40">MV0</font> = <font color="Maroon" title="c39">FV0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:30"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E107:30"/>
<font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">IU</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:30"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E43:30"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E108:30"/><span class="kw">then </span>
<font color="Maroon" title="c39">FV0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E101:30"><span class="lab"><font color="Green" title="E70">A95</font></span></a></span>;<br/>
<a NAME="E109:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A96</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c40">MV0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a>, <a class="txt" href="#E49:30"><span class="lab"><font color="Green" title="E39">A36</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E110:30"/>
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T19" onmouseover="rs('topreal9/T19')" onmouseout="rh()">TOPREAL9:19</a>, <a class="ref" href="xboole_1.html#T76" onmouseover="rs('xboole_1/T76')" onmouseout="rh()">XBOOLE_1:76</a></span>;<br/>
<a NAME="E111:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A97</font></span>: 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:30"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="txt" href="#E90:30"><span class="lab"><font color="Green" title="E65">A90</font></span></a>, <a class="ref" href="funct_1.html#T66" onmouseover="rs('funct_1/T66')" onmouseout="rh()">FUNCT_1:66</a></span>;<br/>
<a NAME="E112:30"/><span class="lab"><font color="Green" title="E73">A98</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:30"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="ref" href="funct_1.html#T62" onmouseover="rs('funct_1/T62')" onmouseout="rh()">FUNCT_1:62</a></span>;<br/>
<a NAME="E113:30"/><span class="lab"><font color="Green" title="E74">A99</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E114:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A100</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c34">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:30"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/>
<a NAME="E115:30"/><span class="lab"><font color="Green" title="E76">A101</font></span>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T18" onmouseover="rs('topreal9/T18')" onmouseout="rh()">TOPREAL9:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c41">ZV0</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a>, <a class="txt" href="#E113:30"><span class="lab"><font color="Green" title="E74">A99</font></span></a></span>;<br/>
<a NAME="E117:30"/><span class="lab"><font color="Green" title="E77">A102</font></span>: 
<font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:30"><span class="lab"><font color="Green" title="E76">A101</font></span></a>, <a class="ref" href="relat_1.html#T120" onmouseover="rs('relat_1/T120')" onmouseout="rh()">RELAT_1:120</a></span>;<br/>
<a NAME="E118:30"/><span class="lab"><font color="Green" title="E78">A103</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E119:30"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c34">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:30"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/>
<span class="kw">then </span> <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) = 
<font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E114:30"><span class="lab"><font color="Green" title="E75">A100</font></span></a>, <a class="ref" href="relat_1.html#T113" onmouseover="rs('relat_1/T113')" onmouseout="rh()">RELAT_1:113</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:30"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="ref" href="funct_1.html#T62" onmouseover="rs('funct_1/T62')" onmouseout="rh()">FUNCT_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E117:30"><span class="lab"><font color="Green" title="E77">A102</font></span></a>, <a class="txt" href="#E112:30"><span class="lab"><font color="Green" title="E73">A98</font></span></a>, <a class="txt" href="#E91:30"><span class="lab"><font color="Green" title="E66">A91</font></span></a>, <a class="txt" href="#E90:30"><span class="lab"><font color="Green" title="E65">A90</font></span></a>, <a class="ref" href="xboole_1.html#T23" onmouseover="rs('xboole_1/T23')" onmouseout="rh()">XBOOLE_1:23</a></span>
;<br/>
<a NAME="E121:30"/><span class="kw">then </span>
<font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:30"><span class="lab"><font color="Green" title="E57">A81</font></span></a>, <a class="txt" href="#E56:30"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="txt" href="#E111:30"><span class="lab"><font color="Green" title="E72">A97</font></span></a>, <a class="ref" href="xboole_1.html#T71" onmouseover="rs('xboole_1/T71')" onmouseout="rh()">XBOOLE_1:71</a></span>;<br/>
<a NAME="E122:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A104</font></span>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">ZV0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a>, <a class="txt" href="#E118:30"><span class="lab"><font color="Green" title="E78">A103</font></span></a></span>;<br/>
<a NAME="E123:30"/><span class="lab"><font color="Green" title="E80">A105</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">ZV0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E113:30"><span class="lab"><font color="Green" title="E74">A99</font></span></a>, <a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c42">GG</font> = <font color="Maroon" title="c34">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c41">ZV0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:30"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="ref" href="jordan24.html#T12" onmouseover="rs('jordan24/T12')" onmouseout="rh()">JORDAN24:12</a>, <a class="txt" href="#E122:30"><span class="lab"><font color="Green" title="E79">A104</font></span></a></span>;<br/>
<a NAME="E125:30"/><span class="lab"><font color="Green" title="E81">A106</font></span>: 
<font color="Maroon" title="c42">GG</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:30"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="ref" href="metrizts.html#T2" onmouseover="rs('metrizts/T2')" onmouseout="rh()">METRIZTS:2</a>, <a class="txt" href="#E123:30"><span class="lab"><font color="Green" title="E80">A105</font></span></a>, <a class="txt" href="#E122:30"><span class="lab"><font color="Green" title="E79">A104</font></span></a></span>;<br/>
<a NAME="E126:30"/><span class="lab"><font color="Green" title="E82">A107</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c40">MV0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c39">FV0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c43">HH</font> = <font color="Maroon" title="c16">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c39">FV0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c39">FV0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">hp</font>,<span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">1 <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2</span>)</span>)</span>)</span>)</span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:30"><span class="lab"><font color="Green" title="E71">A96</font></span></a>, <a class="txt" href="#E68:30"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="ref" href="jordan24.html#T12" onmouseover="rs('jordan24/T12')" onmouseout="rh()">JORDAN24:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c44">GH</font> = <font color="Maroon" title="c42">GG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c43">HH</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c39">FV0</font></span>)</span>,<span class="p1">(<span class="default"><a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:30"><span class="lab"><font color="Green" title="E61">A86</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c39">FV0</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c39">FV0</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> </span><br/>

<a NAME="E129:30"/>
<font color="Maroon" title="c43">HH</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="metrizts.html#T2" onmouseover="rs('metrizts/T2')" onmouseout="rh()">METRIZTS:2</a>, <a class="txt" href="#E109:30"><span class="lab"><font color="Green" title="E71">A96</font></span></a>, <a class="txt" href="#E68:30"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="txt" href="#E126:30"><span class="lab"><font color="Green" title="E82">A107</font></span></a></span>;<br/>
<a NAME="E130:30"/><span class="kw">then </span>
<font color="Maroon" title="c44">GH</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:30"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="txt" href="#E125:30"><span class="lab"><font color="Green" title="E81">A106</font></span></a>, <a class="ref" href="tops_2.html#T57" onmouseover="rs('tops_2/T57')" onmouseout="rh()">TOPS_2:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E131:30"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="mfold_0.html#K3" title="MFOLD_0:func.3">Fr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c39">FV0</font>, <a href="mfold_0.html#K1" title="MFOLD_0:func.1">Tball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,1) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#D1" onmouseover="rs('t_0topsp/D1')" onmouseout="rh()">T_0TOPSP:def 1</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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