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 Th39:
  M // N implies ex X st M c= X & N c= X & X is being_plane
proof
  assume
A1: M // N;
  then N is being_line by AFF_1:36;
  then consider a,b such that
A2: a in N and
  b in N and
  a<>b by AFF_1:19;
A3: M is being_line by A1,AFF_1:36;
  then
A4: ex X st a in X & M c= X & X is being_plane by Th36;
  N=a*M by A1,A3,A2,Def3;
  hence thesis by A3,A4,Th28;
end;
