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

proof

assume between b,p,c ; :: thesis: p out a,b
then ( between c,p,b & between c,p,a ) by A1, Satz3p2;
hence p out a,b by A1, Satz5p2; :: thesis: verum

end;

thus ( p out a,b implies between b,p,c ) :: thesis: verum

proof

assume A2: p out a,b ; :: thesis: between b,p,c
A3: ( between p,a,b implies between b,p,c )

proof

assume between p,a,b ; :: thesis: between b,p,c
then between b,a,p by Satz3p2;
hence between b,p,c by A1, Satz3p7p1; :: thesis: verum

end;

( between p,b,a implies between b,p,c )

proof

assume between p,b,a ; :: thesis: between b,p,c
then between a,b,p by Satz3p2;
hence between b,p,c by A1, Satz3p6p1; :: thesis: verum

end;

hence between b,p,c by A2, A3; :: thesis: verum

end;