let AS be AffinSpace; :: thesis: for c, a, b, d being Element of AS st c in Line (a,b) & d in Line (a,b) & a <> b holds
Line (a,b) c= Line (c,d)
let c, a, b, d be Element of AS; :: thesis: ( c in Line (a,b) & d in Line (a,b) & a <> b implies Line (a,b) c= Line (c,d) )
assume that
A1: c in Line (a,b) and
A2: d in Line (a,b) and
A3: a <> b ; :: thesis: Line (a,b) c= Line (c,d)
A4: LIN a,b,d by A2, Def2;
A5: LIN a,b,c by A1, Def2;

now

let x be set ; :: thesis: ( x in Line (a,b) implies x in Line (c,d) )
assume A6: x in Line (a,b) ; :: thesis: x in Line (c,d)
then reconsider x9 = x as Element of AS ;
LIN a,b,x9 by A6, Def2;
then LIN c,d,x9 by A3, A5, A4, Th17;
hence x in Line (c,d) by Def2; :: thesis: verum

end;

hence Line (a,b) c= Line (c,d) by TARSKI:def 3; :: thesis: verum