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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be  non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">p</font> being  <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> 2</span>)</span>  st <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/>for <font color="Olive" title="b4">rl</font>, <font color="Olive" title="b5">rp</font> being  <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> 2</span>)</span>  st not <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; ( ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b4">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">rp</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b4">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">rp</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) ) &amp;  ex <font color="Olive" title="b6">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b4">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Olive" title="b5">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b6">i</font> ) &amp; not <font color="Olive" title="b4">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Olive" title="b5">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> holds <br/>for <font color="Olive" title="b6">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b6">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">C</font> or  not <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">p</font> be   <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> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies for <font color="Olive" title="b1">rl</font>, <font color="Olive" title="b2">rp</font> being  <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> 2</span>)</span>  st not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; ( ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">rp</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">rp</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) ) &amp;  ex <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b1">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Olive" title="b2">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font> ) &amp; not <font color="Olive" title="b1">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Olive" title="b2">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> holds <br/>for <font color="Olive" title="b3">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b3">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">rl</font>, <font color="Olive" title="b2">rp</font> being  <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> 2</span>)</span>  st not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; ( ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">rp</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">rp</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) ) &amp;  ex <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b1">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Olive" title="b2">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font> ) &amp; not <font color="Olive" title="b1">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Olive" title="b2">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> holds <br/>for <font color="Olive" title="b3">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b3">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">rl</font>, <font color="Maroon" title="c6">rp</font> be   <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> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; ( ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">rp</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">rp</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) ) &amp;  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font> ) &amp; not <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> implies for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
( not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; ( ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">rp</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">rp</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) ) )
 <span class="kw">and </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font> )
 <span class="kw">and </span><br/><a NAME="E4:40"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
not <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E5:40"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:40"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">and </span><br/><a NAME="E8:40"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:40"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E40">A3</font></span></a></span>;<br/>
<a NAME="E10:40"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard5.html#D5" title="GOBOARD5:def.5">GOBOARD5:def 5</a></span>;<br/>
<a NAME="E11:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A10</font></span>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="jordan9.html#T29" title="JORDAN9:th.29">JORDAN9:29</a></span>;<br/>
<a NAME="E12:40"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="jordan9.html#T29" title="JORDAN9:th.29">JORDAN9:29</a></span>;<br/>
<a NAME="E13:40"/>
<font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E14:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A12</font></span>: 
not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="gobrd14.html#T28" title="GOBRD14:th.28">GOBRD14:28</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E50">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1"/>
(  ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="jordan12.html#T31" target="_self" title="JORDAN12:th.31">Th31</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_1"/>
 ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_2"/>
 ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_2"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E16:40"/>
<font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E17:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A15</font></span>: 
not <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="gobrd14.html#T27" title="GOBRD14:th.27">GOBRD14:27</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E52">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E53">A17</font></span>: 
not <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_2_1"/>
(  ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="jordan12.html#T31" target="_self" title="JORDAN12:th.31">Th31</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_2_1_1"/>
 ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_2_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_2_1_2"/>
 ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_2_1_2"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E53">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_3"/><span class="lab"><font color="Green" title="E54">A19</font></span>: 
not <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:40_3"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_3_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="txt" href="#E1:40_3"><span class="lab"><font color="Green" title="E54">A19</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_3_1_1"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_3_1_1_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E2:40_3"><span class="lab"><font color="Green" title="E55">A20</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_3_1_1_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><a NAME="E2:40_3_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_1_1"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_3_1_1_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_3_1_1_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_3_1_1_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_1_1"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_3_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_3_1_2"/><span class="lab"><font color="Green" title="E56">A22</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_3_1_2_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E2:40_3"><span class="lab"><font color="Green" title="E55">A20</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_3_1_2_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_3_1_2_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_1_2"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_3_1_2_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><a NAME="E2:40_3_1_2_1_1_2"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_1_2"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_3_1_2_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_3_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E54">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_4"/><span class="lab"><font color="Green" title="E55">A24</font></span>: 
<font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_4_1"/>
(  ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E1:40_4"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="jordan12.html#E31"><span class="lab"><font color="Green" title="E29">Lm5</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_4_1_1"/>
 ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_4_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_4_1_2"/>
 ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_4_1_2"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E55">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_5"/><span class="lab"><font color="Green" title="E56">A26</font></span>: 
not <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:40_5"/><span class="lab"><font color="Green" title="E57">A27</font></span>: 
<font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_5_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="txt" href="#E1:40_5"><span class="lab"><font color="Green" title="E56">A26</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_5_1_1"/><span class="lab"><font color="Green" title="E58">A28</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_5_1_1_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E2:40_5"><span class="lab"><font color="Green" title="E57">A27</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_5_1_1_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><a NAME="E2:40_5_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_5_1_1"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_5_1_1_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_5_1_1_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_5_1_1_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_5_1_1"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_5_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_5_1_2"/><span class="lab"><font color="Green" title="E58">A29</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_5_1_2_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E2:40_5"><span class="lab"><font color="Green" title="E57">A27</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_5_1_2_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_5_1_2_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_5_1_2"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_5_1_2_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><a NAME="E2:40_5_1_2_1_1_2"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_5_1_2"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_5_1_2_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_5_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E56">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_6"/><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_6_1"/>
(  ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E1:40_6"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="txt" href="jordan12.html#E31"><span class="lab"><font color="Green" title="E29">Lm5</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_6_1_1"/>
 ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_6_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_6_1_2"/>
 ex <font color="Olive" title="b1">C</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> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_6_1_2"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E57">A32</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_7"/><span class="lab"><font color="Green" title="E58">A33</font></span>: 
<font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:40_7"/><span class="lab"><font color="Green" title="E59">A34</font></span>: 
<font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_7_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:40"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="txt" href="#E1:40_7"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_7_1_1"/><span class="lab"><font color="Green" title="E60">A35</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_7_1_1_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E2:40_7"><span class="lab"><font color="Green" title="E59">A34</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_7_1_1_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><a NAME="E2:40_7_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_7_1_1"><span class="lab"><font color="Green" title="E60">A35</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_7_1_1_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_7_1_1_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_7_1_1_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_7_1_1"><span class="lab"><font color="Green" title="E60">A35</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_7_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_7_1_2"/><span class="lab"><font color="Green" title="E60">A36</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_7_1_2_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E2:40_7"><span class="lab"><font color="Green" title="E59">A34</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_7_1_2_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_7_1_2_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_7_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_7_1_2_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><a NAME="E2:40_7_1_2_1_1_2"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_7_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_7_1_2_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_7_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E58">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_8"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_8"/><span class="lab"><font color="Green" title="E59">A38</font></span>: 
<font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:40_8"/><span class="lab"><font color="Green" title="E60">A39</font></span>: 
not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_8_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:40"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="txt" href="#E1:40_8"><span class="lab"><font color="Green" title="E59">A38</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_8_1_1"/><span class="lab"><font color="Green" title="E61">A40</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_8_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_8_1_1_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E2:40_8"><span class="lab"><font color="Green" title="E60">A39</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_8_1_1_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><a NAME="E2:40_8_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_8_1_1"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_8_1_1_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_8_1_1_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_8_1_1_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_8_1_1"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_8_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_8_1_2"/><span class="lab"><font color="Green" title="E61">A41</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_8_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_8_1_2_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E2:40_8"><span class="lab"><font color="Green" title="E60">A39</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_8_1_2_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_8_1_2_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_8_1_2"><span class="lab"><font color="Green" title="E61">A41</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_8_1_2_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><a NAME="E2:40_8_1_2_1_1_2"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_8_1_2"><span class="lab"><font color="Green" title="E61">A41</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_8_1_2_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_8_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_9"/>
( <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1"><span class="kw">suppose </span></a><a NAME="E1:40_9_1"/><span class="lab"><font color="Green" title="E59">A42</font></span>: 
<font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_9_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_9_1_1_1"/>
( <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_9_1_1_1_1"/>
<font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_9_1_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E57">A32</font></span></a>, <a class="txt" href="#E1:40_9_1"><span class="lab"><font color="Green" title="E59">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_9_1_1_1_2"/>
not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_9_1_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="txt" href="#E1:40_9_1"><span class="lab"><font color="Green" title="E59">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_9_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_2"><span class="kw">suppose </span></a><a NAME="E1:40_9_2"/><span class="lab"><font color="Green" title="E59">A43</font></span>: 
not <font color="Maroon" title="c5">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_9_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_9_2_1_1"/>
( <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_9_2_1_1_1"/>
<font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_9_2_1_1_1"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:40"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="txt" href="#E1:40_9_2"><span class="lab"><font color="Green" title="E59">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_9_2_1_1_2"/>
not <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_9_2_1_1_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="txt" href="#E1:40_9_2"><span class="lab"><font color="Green" title="E59">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_9_2"/>
for <font color="Olive" title="b1">C</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> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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