:: PENCIL_1 semantic presentation
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theorem Th1: :: PENCIL_1:1
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theorem Th2: :: PENCIL_1:2
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for
X being
set holds
( 2
c= Card X iff ex
x,
y being
set st
(
x in X &
y in X &
x <> y ) )
theorem Th3: :: PENCIL_1:3
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theorem Th4: :: PENCIL_1:4
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theorem Th5: :: PENCIL_1:5
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for
X being
set holds
( 3
c= Card X iff ex
x,
y,
z being
set st
(
x in X &
y in X &
z in X &
x <> y &
x <> z &
y <> z ) )
theorem Th6: :: PENCIL_1:6
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:: deftheorem Def1 defines are_collinear PENCIL_1:def 1 :
:: deftheorem Def2 defines closed_under_lines PENCIL_1:def 2 :
:: deftheorem Def3 defines strong PENCIL_1:def 3 :
:: deftheorem Def4 defines void PENCIL_1:def 4 :
:: deftheorem Def5 defines degenerated PENCIL_1:def 5 :
:: deftheorem Def6 defines with_non_trivial_blocks PENCIL_1:def 6 :
:: deftheorem Def7 defines identifying_close_blocks PENCIL_1:def 7 :
:: deftheorem Def8 defines truly-partial PENCIL_1:def 8 :
:: deftheorem Def9 defines without_isolated_points PENCIL_1:def 9 :
:: deftheorem defines connected PENCIL_1:def 10 :
:: deftheorem Def11 defines strongly_connected PENCIL_1:def 11 :
theorem Th7: :: PENCIL_1:7
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Lm1:
ex S being TopStruct st
( S is strict & not S is empty & not S is void & not S is degenerated & not S is truly-partial & S is with_non_trivial_blocks & S is identifying_close_blocks & S is without_isolated_points )
theorem Th8: :: PENCIL_1:8
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Lm2:
ex S being TopStruct st
( S is strict & not S is empty & not S is void & not S is degenerated & S is truly-partial & S is with_non_trivial_blocks & S is identifying_close_blocks & S is without_isolated_points )
:: deftheorem defines are_collinear PENCIL_1:def 12 :
:: deftheorem Def13 defines TopStruct-yielding PENCIL_1:def 13 :
:: deftheorem Def14 defines non-void-yielding PENCIL_1:def 14 :
:: deftheorem defines non-void-yielding PENCIL_1:def 15 :
:: deftheorem Def16 defines trivial-yielding PENCIL_1:def 16 :
:: deftheorem Def17 defines non-Trivial-yielding PENCIL_1:def 17 :
:: deftheorem defines non-Trivial-yielding PENCIL_1:def 18 :
:: deftheorem Def19 defines PLS-yielding PENCIL_1:def 19 :
:: deftheorem Def20 defines Segre-like PENCIL_1:def 20 :
theorem :: PENCIL_1:9
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theorem :: PENCIL_1:10
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theorem Th11: :: PENCIL_1:11
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:: deftheorem Def21 defines indx PENCIL_1:def 21 :
theorem Th12: :: PENCIL_1:12
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theorem Th13: :: PENCIL_1:13
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:: deftheorem Def22 defines Segre_Blocks PENCIL_1:def 22 :
:: deftheorem defines Segre_Product PENCIL_1:def 23 :
theorem Th14: :: PENCIL_1:14
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theorem Th15: :: PENCIL_1:15
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theorem Th16: :: PENCIL_1:16
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theorem Th17: :: PENCIL_1:17
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theorem Th18: :: PENCIL_1:18
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theorem :: PENCIL_1:19
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theorem Th20: :: PENCIL_1:20
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theorem Th21: :: PENCIL_1:21
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theorem :: PENCIL_1:22
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theorem Th23: :: PENCIL_1:23
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theorem Th24: :: PENCIL_1:24
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theorem Th25: :: PENCIL_1:25
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theorem Th26: :: PENCIL_1:26
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theorem Th27: :: PENCIL_1:27
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theorem :: PENCIL_1:28
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theorem :: PENCIL_1:29
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