reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem
  A is being_line & X is being_plane & A '||' X implies a*A c= a+X
proof
  assume that
A1: A is being_line and
A2: X is being_plane and
A3: A '||' X;
A4: X '||' a+X & a in a+ X by A2,Def6;
  consider N 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 thesis by A5,A4;
end;
