:: CONMETR semantic presentation

begin

definition
let X be ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) ;
attr X is satisfying_OPAP means :: CONMETR:def 1
for o, a1, a2, a3, b1, b2, b3 being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for M, N being ( ( ) ( ) Subset of ) st o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & not a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ N : ( ( ) ( ) Subset of ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
attr X is satisfying_PAP means :: CONMETR:def 2
for o, a1, a2, a3, b1, b2, b3 being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for M, N being ( ( ) ( ) Subset of ) st M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) is being_line & N : ( ( ) ( ) Subset of ) is being_line & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & not a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
attr X is satisfying_MH1 means :: CONMETR:def 3
for a1, a2, a3, a4, b1, b2, b3, b4 being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for M, N being ( ( ) ( ) Subset of ) st M : ( ( ) ( ) Subset of ) _|_ N : ( ( ) ( ) Subset of ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
attr X is satisfying_MH2 means :: CONMETR:def 4
for a1, a2, a3, a4, b1, b2, b3, b4 being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for M, N being ( ( ) ( ) Subset of ) st M : ( ( ) ( ) Subset of ) _|_ N : ( ( ) ( ) Subset of ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) _|_ b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
attr X is satisfying_TDES means :: CONMETR:def 5
for o, a, a1, b, b1, c, c1 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> c1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & not LIN b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & not LIN b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & LIN o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & LIN o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & LIN o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // o : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
attr X is satisfying_SCH means :: CONMETR:def 6
for a1, a2, a3, a4, b1, b2, b3, b4 being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for M, N being ( ( ) ( ) Subset of ) st M : ( ( ) ( ) Subset of ) is being_line & N : ( ( ) ( ) Subset of ) is being_line & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
attr X is satisfying_OSCH means :: CONMETR:def 7
for a1, a2, a3, a4, b1, b2, b3, b4 being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for M, N being ( ( ) ( ) Subset of ) st M : ( ( ) ( ) Subset of ) _|_ N : ( ( ) ( ) Subset of ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & not b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in N : ( ( ) ( ) Subset of ) & a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b2 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
a3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b3 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b4 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
attr X is satisfying_des means :: CONMETR:def 8
for a, a1, b, b1, c, c1 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st not LIN a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & not LIN a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // b1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c1 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
end;

theorem :: CONMETR:1
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) ex a, b, c being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st
( LIN a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) ;

theorem :: CONMETR:2
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl)
for a, b being ( ( ) ( ) Element of ( ( ) ( ) set ) ) ex c being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st
( LIN a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) <> c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) ;

theorem :: CONMETR:3
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl)
for A being ( ( ) ( ) Subset of )
for a being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st A : ( ( ) ( ) Subset of ) is being_line holds
ex K being ( ( ) ( ) Subset of ) st
( a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in K : ( ( ) ( ) Subset of ) & A : ( ( ) ( ) Subset of ) _|_ K : ( ( ) ( ) Subset of ) ) ;

theorem :: CONMETR:4
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) is being_line & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in A : ( ( ) ( ) Subset of ) & b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in A : ( ( ) ( ) Subset of ) & c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in A : ( ( ) ( ) Subset of ) holds
LIN a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;

theorem :: CONMETR:5
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl)
for a, b being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for A, M being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) is being_line & M : ( ( ) ( ) Subset of ) is being_line & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in A : ( ( ) ( ) Subset of ) & b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in A : ( ( ) ( ) Subset of ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & not a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) = b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
A : ( ( ) ( ) Subset of ) = M : ( ( ) ( ) Subset of ) ;

theorem :: CONMETR:6
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for M being ( ( ) ( ) Subset of )
for M9 being ( ( ) ( ) Subset of )
for c9, d9 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) = c9 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & d : ( ( ) ( ) Element of ( ( ) ( ) set ) ) = d9 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & M : ( ( ) ( ) Subset of ) = M9 : ( ( ) ( ) Subset of ) & a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in M : ( ( ) ( ) Subset of ) & c9 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,d9 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // M9 : ( ( ) ( ) Subset of ) holds
c : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( ) set ) ) // a : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;

theorem :: CONMETR:7
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_TDES holds
Af X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) : ( ( strict ) ( V41() V46() strict V103() V104() ) AffinStruct ) is Moufangian ;

theorem :: CONMETR:8
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st Af X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) : ( ( strict ) ( V41() V46() strict V103() V104() ) AffinStruct ) is translational holds
X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_des ;

theorem :: CONMETR:9
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_MH1 holds
X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_OSCH ;

theorem :: CONMETR:10
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_MH2 holds
X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_OSCH ;

theorem :: CONMETR:11
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_AH holds
X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_TDES ;

theorem :: CONMETR:12
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_OSCH & X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_TDES holds
X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_SCH ;

theorem :: CONMETR:13
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_OPAP & X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_DES holds
X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_PAP ;

theorem :: CONMETR:14
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_MH1 & X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_MH2 holds
X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_OPAP ;

theorem :: CONMETR:15
for X being ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) st X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_3H holds
X : ( ( V41() OrtAfPl-like ) ( V41() OrtAfSp-like OrtAfPl-like ) OrtAfPl) is satisfying_OPAP ;