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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_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="E1:11_1_1_1"/>
( <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:11_1_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_1_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> 
 ; <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="11_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:11_1_1_1_2"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
not <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> </span><br/><div class="add"><a NAME="E2:11_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A2</font></span>: 
not <font color="Maroon" title="c2">TM2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c3">metr2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TM2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TM2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:11_1_1_1_2"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c3">metr2</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TM2</font>
 <span class="kw">and </span><br/><a NAME="E5:11_1_1_1_2"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TM2</font>,<font color="Maroon" title="c3">metr2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">TM2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#D9" title="PCOMPS_1:def.9">PCOMPS_1:def 9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">tm2</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TM2</font>,<font color="Maroon" title="c3">metr2</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">metr1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E7:11_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c5">metr1</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM1</font>
 <span class="kw">and </span><br/><a NAME="E8:11_1_1_1_2"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
 <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c5">metr1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#D9" title="PCOMPS_1:def.9">PCOMPS_1:def 9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">tm1</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c5">metr1</font>;<br/><a NAME="E9:11_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
not <font color="Maroon" title="c1">TM1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_1_2"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">tm1</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c5">metr1</font>, <font color="Maroon" title="c8">tm2</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TM2</font>,<font color="Maroon" title="c3">metr2</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1_1_1_2"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E4:11_1_1_1_2"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E2:11_1_1_1_2"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="pcomps_1.html#T40" title="PCOMPS_1:th.40">PCOMPS_1:40</a></span>;<br/><a NAME="E11:11_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TM2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">TM2</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c8">tm2</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c8">tm2</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_1_1_2"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E5:11_1_1_1_2"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E2:11_1_1_1_2"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></span>;<br/><a NAME="E12:11_1_1_1_2"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c7">tm1</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c7">tm1</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E7:11_1_1_1_2"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E8:11_1_1_1_2"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></span>;<br/><span class="kw">then </span><span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> = 
<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="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c7">tm1</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c8">tm2</font></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="#E11:11_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="waybel29.html#T10" title="WAYBEL29:th.10">WAYBEL29:10</a></span>
<br/>.= 
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <font color="Maroon" title="c7">tm1</font>,<font color="Maroon" title="c8">tm2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#T25" title="TOPREAL7:th.25">TOPREAL7:25</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:11_1_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">TM1</font>,<font color="Maroon" title="c2">TM2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> 
 ; <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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