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 in A & a in X & A '||' X
  implies A c= X
proof
  assume that
A1: A is being_line and
A2: X is being_plane and
A3: a in A and
A4: a in X and
A5: A '||' X;
  consider N such that
A6: N c= X and
A7: A // N or N // A by A1,A2,A5,Th41;
A8: N is being_line by A7,AFF_1:36;
  A=a*A by A1,A3,Lm8
    .= a*N by A7,Th32;
  hence thesis by A2,A4,A6,A8,Th28;
end;
