reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;
reserve x,y,z,t,u,w for Element of AS;
reserve K,X,Y,Z,X9,Y9 for Subset of AS;
reserve a,b,c,d,p,q,r,p9 for POINT of IncProjSp_of(AS);
reserve A for LINE of IncProjSp_of(AS);

theorem Th31:
  X is being_line & Y is being_plane & X c= Y & a=LDir(X) & A=[
  PDir(Y),2] implies a on A
proof
  assume that
A1: X is being_line and
A2: Y is being_plane and
A3: X c= Y and
A4: a=LDir(X) and
A5: A=[PDir(Y),2];
  X '||' Y by A1,A2,A3,AFF_4:42;
  hence thesis by A1,A2,A4,A5,Th29;
end;
