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 Th41:
  M is being_line & X is being_plane implies (M '||' X iff ex N st
  N c= X & (M // N or N // M) )
proof
  assume that
A1: M is being_line and
A2: X is being_plane;
A3: now
    given N such that
A4: N c= X and
A5: M // N or N // M;
    now
      let a,A;
      assume that
A6:   a in X and
A7:   A is being_line and
A8:   A c= M;
      A=M by A1,A7,A8,Th33;
      then M // a*A by A7,Def3;
      then
A9:   N // a*A by A5,AFF_1:44;
      a in a*A by A7,Def3;
      hence a*A c= X by A2,A4,A6,A9,Th23;
    end;
    hence M '||' X;
  end;
  now
    consider a,b,c such that
A10: a in X and
    b in X and
    c in X and
    not LIN a,b,c by A2,Th34;
    assume
A11: M '||' X;
    take N=a*M;
    thus N c= X by A1,A11,A10;
    thus M // N or N // M by A1,Def3;
  end;
  hence thesis by A3;
end;
