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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st (  for <font color="Olive" title="b2">fi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span><br/>  for <font color="Olive" title="b3">fj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span> holds  <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">fi</font>,<font color="Olive" title="b3">fj</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">fj</font> ) holds <br/> for <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b3">fi</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span><br/>  for <font color="Olive" title="b4">fj</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span><br/>  for <font color="Olive" title="b5">fij</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#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st <font color="Olive" title="b5">fij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b4">fj</font> holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b3">fi</font>,<font color="Olive" title="b2">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b4">fj</font>,<font color="Olive" title="b2">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b5">fij</font>,<font color="Olive" title="b2">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">fi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span><br/>  for <font color="Olive" title="b2">fj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span> holds  <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">fi</font>,<font color="Olive" title="b2">fj</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">fj</font> ) implies  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b2">fi</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span><br/>  for <font color="Olive" title="b3">fj</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span><br/>  for <font color="Olive" title="b4">fij</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#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st <font color="Olive" title="b4">fij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">fj</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b2">fi</font>,<font color="Olive" title="b1">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b3">fj</font>,<font color="Olive" title="b1">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b4">fij</font>,<font color="Olive" title="b1">r</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 for <font color="Olive" title="b1">fi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span><br/>  for <font color="Olive" title="b2">fj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span> holds  <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">fi</font>,<font color="Olive" title="b2">fj</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">fj</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b2">fi</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span><br/>  for <font color="Olive" title="b3">fj</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span><br/>  for <font color="Olive" title="b4">fij</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#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st <font color="Olive" title="b4">fij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">fj</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b2">fi</font>,<font color="Olive" title="b1">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b3">fj</font>,<font color="Olive" title="b1">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b4">fij</font>,<font color="Olive" title="b1">r</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">fi</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span><br/>  for <font color="Olive" title="b2">fj</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span><br/>  for <font color="Olive" title="b3">fij</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#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">fij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">fj</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b1">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b2">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b3">fij</font>,<font color="Maroon" title="c4">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">fi</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">fj</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span><br/>  for <font color="Olive" title="b2">fij</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#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">fij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">fj</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b1">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b2">fij</font>,<font color="Maroon" title="c4">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">fj</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">fij</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#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">fij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">fj</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Olive" title="b1">fij</font>,<font color="Maroon" title="c4">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">fij</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#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">fij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">fj</font> implies <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E2:35"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c7">fij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">fj</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">Ii</font> =  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">Ij</font> =  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Iij</font> =  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>);<br/>
<a NAME="E3:35"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D4" title="EUCLID_9:def.4">EUCLID_9:def 4</a></span>;<br/>
<a NAME="E4:35"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="simplex2.html#E29"><span class="lab"><font color="Green" title="E21">Lm3</font></span></a></span>;<br/>
<a NAME="E5:35"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="simplex2.html#E29"><span class="lab"><font color="Green" title="E21">Lm3</font></span></a></span>;<br/>
<a NAME="E6:35"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D4" title="EUCLID_9:def.4">EUCLID_9:def 4</a></span>;<br/>
<a NAME="E7:35"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="rlaffin3.html#T1" title="RLAFFIN3:th.1">RLAFFIN3:1</a></span>;<br/>
<a NAME="E8:35"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:35_1"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">xi</font>, <font color="Maroon" title="c14">xj</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:35_1"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c13">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:35_1"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c14">xj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:35_1"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">xi</font>,<font color="Maroon" title="c14">xj</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">xi</font> = <font color="Maroon" title="c13">xi</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">xj</font> = <font color="Maroon" title="c14">xj</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E8:35_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a></span>;<br/>
<font color="Maroon" title="c11">y</font> = 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">xi</font>,<font color="Maroon" title="c16">xj</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="binop_1.html#D1" title="BINOP_1:def.1">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c15">xi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">xj</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:35_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E8:35_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="rlaffin3.html#T2" title="RLAFFIN3:th.2">RLAFFIN3:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:35"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> or <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p1</font>, <font color="Maroon" title="c13">p2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:35_2"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">p2</font>
 <span class="kw">and </span><br/><a NAME="E4:35_2"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
( <font color="Maroon" title="c12">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> &amp; <font color="Maroon" title="c13">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="rlaffin3.html#T2" title="RLAFFIN3:th.2">RLAFFIN3:2</a></span>;<br/>
<a NAME="E5:35_2"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E6:35_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E7:35_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E8:35_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c11">y</font> = 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E3:35_2"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E33">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" title="BINOP_1:def.1">BINOP_1:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:35_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_2"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E8:35_2"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:35"/>
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D4" title="EUCLID_9:def.4">EUCLID_9:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:35"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c5">fi</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c6">fj</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c7">fij</font>,<font color="Maroon" title="c4">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E9:35"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
