let S be satisfying_CongruenceIdentity satisfying_SegmentConstruction satisfying_BetweennessIdentity satisfying_Pasch TarskiGeometryStruct ; :: thesis: for a, b, c, d being POINT of S st between a,b,d & between b,c,d holds
between a,b,c
let a, b, c, d be POINT of S; :: thesis: ( between a,b,d & between b,c,d implies between a,b,c )
assume ( between a,b,d & between b,c,d ) ; :: thesis: between a,b,c
then consider x being POINT of S such that
A1: ( between b,x,b & between c,x,a ) by GTARSKI1:def 11;
b = x by A1, GTARSKI1:def 10;
hence between a,b,c by A1, Satz3p2; :: thesis: verum