let AS be AffinSpace; :: thesis: for a being Element of AS
for A being being_line Subset of AS ex C being Subset of AS st
( a in C & A // C )
let a be Element of AS; :: thesis: for A being being_line Subset of AS ex C being Subset of AS st
( a in C & A // C )
let A be being_line Subset of AS; :: thesis: ex C being Subset of AS st
( a in C & A // C )
consider p, q being Element of AS such that
A1: p <> q and
A2: A = Line (p,q) by Def3;
consider b being Element of AS such that
A3: p,q // a,b and
A4: a <> b by DIRAF:40;
set C = Line (a,b);
A5: a in Line (a,b) by Th14;
A // Line (a,b) by A1, A2, A3, A4, Th36;
hence ex C being Subset of AS st
( a in C & A // C ) by A5; :: thesis: verum