let AS be AffinSpace; :: thesis: for a being Element of AS
for A, X being Subset of AS st A is being_line & X is being_plane & A '||' X holds
a * A c= a + X
let a be Element of AS; :: thesis: for A, X being Subset of AS st A is being_line & X is being_plane & A '||' X holds
a * A c= a + X
let A, X be Subset of AS; :: thesis: ( A is being_line & X is being_plane & A '||' X implies a * A c= a + X )
assume that
A1: A is being_line and
A2: X is being_plane and
A3: A '||' X ; :: thesis: a * A c= a + X
A4: ( X '||' a + X & a in a + X ) by A2, Def6;
consider N being Subset of AS such that
A5: N c= X and
A6: ( A // N or N // A ) by A1, A2, A3, Th41;
( a * A = a * N & N is being_line ) by A6, Th32, AFF_1:36;
hence a * A c= a + X by A5, A4; :: thesis: verum