:: CONMETR semantic presentation
definition
let c1 be
OrtAfPl;
attr a1 is
satisfying_OPAP means :
Def1:
:: CONMETR:def 1
for
b1,
b2,
b3,
b4,
b5,
b6,
b7 being
Element of
a1for
b8,
b9 being
Subset of
a1 st
b1 in b8 &
b2 in b8 &
b3 in b8 &
b4 in b8 &
b1 in b9 &
b5 in b9 &
b6 in b9 &
b7 in b9 & not
b6 in b8 & not
b4 in b9 &
b8 _|_ b9 &
b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b1 <> b6 &
b1 <> b7 &
b4,
b6 // b3,
b5 &
b4,
b7 // b2,
b5 holds
b2,
b6 // b3,
b7;
attr a1 is
satisfying_PAP means :: CONMETR:def 2
for
b1,
b2,
b3,
b4,
b5,
b6,
b7 being
Element of
a1for
b8,
b9 being
Subset of
a1 st
b8 is_line &
b9 is_line &
b1 in b8 &
b2 in b8 &
b3 in b8 &
b4 in b8 &
b1 in b9 &
b5 in b9 &
b6 in b9 &
b7 in b9 & not
b6 in b8 & not
b4 in b9 &
b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b1 <> b6 &
b1 <> b7 &
b4,
b6 // b3,
b5 &
b4,
b7 // b2,
b5 holds
b2,
b6 // b3,
b7;
attr a1 is
satisfying_MH1 means :
Def3:
:: CONMETR:def 3
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
Element of
a1for
b9,
b10 being
Subset of
a1 st
b9 _|_ b10 &
b1 in b9 &
b3 in b9 &
b5 in b9 &
b7 in b9 &
b2 in b10 &
b4 in b10 &
b6 in b10 &
b8 in b10 & not
b2 in b9 & not
b4 in b9 &
b1,
b2 _|_ b5,
b6 &
b2,
b3 _|_ b6,
b7 &
b3,
b4 _|_ b7,
b8 holds
b1,
b4 _|_ b5,
b8;
attr a1 is
satisfying_MH2 means :
Def4:
:: CONMETR:def 4
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
Element of
a1for
b9,
b10 being
Subset of
a1 st
b9 _|_ b10 &
b1 in b9 &
b3 in b9 &
b6 in b9 &
b8 in b9 &
b2 in b10 &
b4 in b10 &
b5 in b10 &
b7 in b10 & not
b2 in b9 & not
b4 in b9 &
b1,
b2 _|_ b5,
b6 &
b2,
b3 _|_ b6,
b7 &
b3,
b4 _|_ b7,
b8 holds
b1,
b4 _|_ b5,
b8;
attr a1 is
satisfying_TDES means :
Def5:
:: CONMETR:def 5
for
b1,
b2,
b3,
b4,
b5,
b6,
b7 being
Element of
a1 st
b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b1 <> b6 &
b1 <> b7 & not
LIN b4,
b5,
b2 & not
LIN b4,
b5,
b6 &
LIN b1,
b2,
b3 &
LIN b1,
b4,
b5 &
LIN b1,
b6,
b7 &
b2,
b4 // b3,
b5 &
b2,
b4 // b1,
b6 &
b4,
b6 // b5,
b7 holds
b2,
b6 // b3,
b7;
attr a1 is
satisfying_SCH means :: CONMETR:def 6
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
Element of
a1for
b9,
b10 being
Subset of
a1 st
b9 is_line &
b10 is_line &
b1 in b9 &
b3 in b9 &
b5 in b9 &
b7 in b9 &
b2 in b10 &
b4 in b10 &
b6 in b10 &
b8 in b10 & not
b4 in b9 & not
b2 in b9 & not
b6 in b9 & not
b8 in b9 & not
b1 in b10 & not
b3 in b10 & not
b5 in b10 & not
b7 in b10 &
b3,
b2 // b7,
b6 &
b2,
b1 // b6,
b5 &
b1,
b4 // b5,
b8 holds
b3,
b4 // b7,
b8;
attr a1 is
satisfying_OSCH means :
Def7:
:: CONMETR:def 7
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
Element of
a1for
b9,
b10 being
Subset of
a1 st
b9 _|_ b10 &
b1 in b9 &
b3 in b9 &
b5 in b9 &
b7 in b9 &
b2 in b10 &
b4 in b10 &
b6 in b10 &
b8 in b10 & not
b4 in b9 & not
b2 in b9 & not
b6 in b9 & not
b8 in b9 & not
b1 in b10 & not
b3 in b10 & not
b5 in b10 & not
b7 in b10 &
b3,
b2 // b7,
b6 &
b2,
b1 // b6,
b5 &
b1,
b4 // b5,
b8 holds
b3,
b4 // b7,
b8;
attr a1 is
satisfying_des means :
Def8:
:: CONMETR:def 8
for
b1,
b2,
b3,
b4,
b5,
b6 being
Element of
a1 st not
LIN b1,
b2,
b3 & not
LIN b1,
b2,
b5 &
b1,
b2 // b3,
b4 &
b1,
b2 // b5,
b6 &
b1,
b3 // b2,
b4 &
b1,
b5 // b2,
b6 holds
b3,
b5 // b4,
b6;
end;
:: deftheorem Def1 defines satisfying_OPAP CONMETR:def 1 :
for
b1 being
OrtAfPl holds
(
b1 is
satisfying_OPAP iff for
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
Element of
b1for
b9,
b10 being
Subset of
b1 st
b2 in b9 &
b3 in b9 &
b4 in b9 &
b5 in b9 &
b2 in b10 &
b6 in b10 &
b7 in b10 &
b8 in b10 & not
b7 in b9 & not
b5 in b10 &
b9 _|_ b10 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b2 <> b6 &
b2 <> b7 &
b2 <> b8 &
b5,
b7 // b4,
b6 &
b5,
b8 // b3,
b6 holds
b3,
b7 // b4,
b8 );
:: deftheorem Def2 defines satisfying_PAP CONMETR:def 2 :
for
b1 being
OrtAfPl holds
(
b1 is
satisfying_PAP iff for
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
Element of
b1for
b9,
b10 being
Subset of
b1 st
b9 is_line &
b10 is_line &
b2 in b9 &
b3 in b9 &
b4 in b9 &
b5 in b9 &
b2 in b10 &
b6 in b10 &
b7 in b10 &
b8 in b10 & not
b7 in b9 & not
b5 in b10 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b2 <> b6 &
b2 <> b7 &
b2 <> b8 &
b5,
b7 // b4,
b6 &
b5,
b8 // b3,
b6 holds
b3,
b7 // b4,
b8 );
:: deftheorem Def3 defines satisfying_MH1 CONMETR:def 3 :
for
b1 being
OrtAfPl holds
(
b1 is
satisfying_MH1 iff for
b2,
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
Element of
b1for
b10,
b11 being
Subset of
b1 st
b10 _|_ b11 &
b2 in b10 &
b4 in b10 &
b6 in b10 &
b8 in b10 &
b3 in b11 &
b5 in b11 &
b7 in b11 &
b9 in b11 & not
b3 in b10 & not
b5 in b10 &
b2,
b3 _|_ b6,
b7 &
b3,
b4 _|_ b7,
b8 &
b4,
b5 _|_ b8,
b9 holds
b2,
b5 _|_ b6,
b9 );
:: deftheorem Def4 defines satisfying_MH2 CONMETR:def 4 :
for
b1 being
OrtAfPl holds
(
b1 is
satisfying_MH2 iff for
b2,
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
Element of
b1for
b10,
b11 being
Subset of
b1 st
b10 _|_ b11 &
b2 in b10 &
b4 in b10 &
b7 in b10 &
b9 in b10 &
b3 in b11 &
b5 in b11 &
b6 in b11 &
b8 in b11 & not
b3 in b10 & not
b5 in b10 &
b2,
b3 _|_ b6,
b7 &
b3,
b4 _|_ b7,
b8 &
b4,
b5 _|_ b8,
b9 holds
b2,
b5 _|_ b6,
b9 );
:: deftheorem Def5 defines satisfying_TDES CONMETR:def 5 :
for
b1 being
OrtAfPl holds
(
b1 is
satisfying_TDES iff for
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
Element of
b1 st
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b2 <> b6 &
b2 <> b7 &
b2 <> b8 & not
LIN b5,
b6,
b3 & not
LIN b5,
b6,
b7 &
LIN b2,
b3,
b4 &
LIN b2,
b5,
b6 &
LIN b2,
b7,
b8 &
b3,
b5 // b4,
b6 &
b3,
b5 // b2,
b7 &
b5,
b7 // b6,
b8 holds
b3,
b7 // b4,
b8 );
:: deftheorem Def6 defines satisfying_SCH CONMETR:def 6 :
for
b1 being
OrtAfPl holds
(
b1 is
satisfying_SCH iff for
b2,
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
Element of
b1for
b10,
b11 being
Subset of
b1 st
b10 is_line &
b11 is_line &
b2 in b10 &
b4 in b10 &
b6 in b10 &
b8 in b10 &
b3 in b11 &
b5 in b11 &
b7 in b11 &
b9 in b11 & not
b5 in b10 & not
b3 in b10 & not
b7 in b10 & not
b9 in b10 & not
b2 in b11 & not
b4 in b11 & not
b6 in b11 & not
b8 in b11 &
b4,
b3 // b8,
b7 &
b3,
b2 // b7,
b6 &
b2,
b5 // b6,
b9 holds
b4,
b5 // b8,
b9 );
:: deftheorem Def7 defines satisfying_OSCH CONMETR:def 7 :
for
b1 being
OrtAfPl holds
(
b1 is
satisfying_OSCH iff for
b2,
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
Element of
b1for
b10,
b11 being
Subset of
b1 st
b10 _|_ b11 &
b2 in b10 &
b4 in b10 &
b6 in b10 &
b8 in b10 &
b3 in b11 &
b5 in b11 &
b7 in b11 &
b9 in b11 & not
b5 in b10 & not
b3 in b10 & not
b7 in b10 & not
b9 in b10 & not
b2 in b11 & not
b4 in b11 & not
b6 in b11 & not
b8 in b11 &
b4,
b3 // b8,
b7 &
b3,
b2 // b7,
b6 &
b2,
b5 // b6,
b9 holds
b4,
b5 // b8,
b9 );
:: deftheorem Def8 defines satisfying_des CONMETR:def 8 :
for
b1 being
OrtAfPl holds
(
b1 is
satisfying_des iff for
b2,
b3,
b4,
b5,
b6,
b7 being
Element of
b1 st not
LIN b2,
b3,
b4 & not
LIN b2,
b3,
b6 &
b2,
b3 // b4,
b5 &
b2,
b3 // b6,
b7 &
b2,
b4 // b3,
b5 &
b2,
b6 // b3,
b7 holds
b4,
b6 // b5,
b7 );
theorem Th1: :: CONMETR:1
theorem Th2: :: CONMETR:2
theorem Th3: :: CONMETR:3
theorem Th4: :: CONMETR:4
theorem Th5: :: CONMETR:5
theorem Th6: :: CONMETR:6
theorem Th7: :: CONMETR:7
theorem Th8: :: CONMETR:8
theorem Th9: :: CONMETR:9
theorem Th10: :: CONMETR:10
theorem Th11: :: CONMETR:11
theorem Th12: :: CONMETR:12
theorem Th13: :: CONMETR:13
theorem Th14: :: CONMETR:14
theorem Th15: :: CONMETR:15