let AS be AffinSpace; :: thesis: for a, b being Element of AS
for A being Subset of AS st A is being_line & a in A & b in A & a <> b holds
A = Line (a,b)
let a, b be Element of AS; :: thesis: for A being Subset of AS st A is being_line & a in A & b in A & a <> b holds
A = Line (a,b)
let A be Subset of AS; :: thesis: ( A is being_line & a in A & b in A & a <> b implies A = Line (a,b) )
assume that
A1: A is being_line and
A2: a in A and
A3: b in A and
A4: a <> b ; :: thesis: A = Line (a,b)
A5: ex p, q being Element of AS st
( p <> q & A = Line (p,q) ) by A1;
then A6: A c= Line (a,b) by A2, A3, Th16;
Line (a,b) c= A by A2, A3, A4, A5, Th15;
hence A = Line (a,b) by A6, XBOOLE_0:def 10; :: thesis: verum