let S be satisfying_Tarski-model TarskiGeometryStruct ; :: thesis: for a, b, c, d, p being POINT of S st between4 a,b,c,d & between c,d,p holds
( between c,d,p & ( c <> d implies between5 a,b,c,d,p ) )
let a, b, c, d, p be POINT of S; :: thesis: ( between4 a,b,c,d & between c,d,p implies ( between c,d,p & ( c <> d implies between5 a,b,c,d,p ) ) )
assume A1: ( between4 a,b,c,d & between c,d,p ) ; :: thesis: ( between c,d,p & ( c <> d implies between5 a,b,c,d,p ) )
hence between c,d,p ; :: thesis: ( c <> d implies between5 a,b,c,d,p )

hereby :: thesis: verum

assume c <> d ; :: thesis: between5 a,b,c,d,p
then ( between b,c,p & between b,d,p & between a,c,p & between a,d,p ) by A1, Satz3p7p1, Satz3p7p2;
hence between5 a,b,c,d,p by A1, Satz3p7p2; :: thesis: verum

end;