let AS be AffinSpace; :: thesis: for X, Y being Subset of AS
for a being Element of the Points of (ProjHorizon AS)
for A being Element of the Lines of (ProjHorizon AS) st a = LDir X & A = PDir Y & X is being_line & Y is being_plane holds
( a on A iff X '||' Y )
let X, Y be Subset of AS; :: thesis: for a being Element of the Points of (ProjHorizon AS)
for A being Element of the Lines of (ProjHorizon AS) st a = LDir X & A = PDir Y & X is being_line & Y is being_plane holds
( a on A iff X '||' Y )
let a be Element of the Points of (ProjHorizon AS); :: thesis: for A being Element of the Lines of (ProjHorizon AS) st a = LDir X & A = PDir Y & X is being_line & Y is being_plane holds
( a on A iff X '||' Y )
let A be Element of the Lines of (ProjHorizon AS); :: thesis: ( a = LDir X & A = PDir Y & X is being_line & Y is being_plane implies ( a on A iff X '||' Y ) )
assume that
A1: ( a = LDir X & A = PDir Y ) and
A2: ( X is being_line & Y is being_plane ) ; :: thesis: ( a on A iff X '||' Y )

A3: now

assume X '||' Y ; :: thesis: a on A
then [a,A] in Inc_of_Dir AS by A1, A2, Def12;
hence a on A by INCSP_1:def 1; :: thesis: verum

end;

now

assume a on A ; :: thesis: X '||' Y
then [a,A] in the Inc of (ProjHorizon AS) by INCSP_1:def 1;
then consider X', Y' being Subset of AS such that
A4: ( a = LDir X' & A = PDir Y' & X' is being_line & Y' is being_plane & X' '||' Y' ) by Def12;
A5: ( X // X' & Y' '||' Y ) by A1, A2, A4, Th11, Th13;
then X '||' Y' by A4, AFF_4:56;
hence X '||' Y by A4, A5, AFF_4:59, AFF_4:60; :: thesis: verum

end;

hence ( a on A iff X '||' Y ) by A3; :: thesis: verum