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 a,d,p holds
( between5 a,b,c,d,p & ( a <> 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 a,d,p implies ( between5 a,b,c,d,p & ( a <> d implies between5 a,b,c,d,p ) ) )
assume A1: ( between4 a,b,c,d & between a,d,p ) ; :: thesis: ( between5 a,b,c,d,p & ( a <> d implies between5 a,b,c,d,p ) )
then ( between a,b,p & between a,c,p & between a,d,p & between c,d,p & between b,d,p ) by Satz3p6p1, Satz3p6p2;
hence ( between5 a,b,c,d,p & ( a <> d implies between5 a,b,c,d,p ) ) by A1, Satz3p6p1; :: thesis: verum