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 Th32:
  Y is being_plane & X c= Y & X9 // X & a=LDir(X9) & A=[PDir(Y),2]
  implies a on A
proof
  assume that
A1: Y is being_plane and
A2: X c= Y and
A3: X9 // X and
A4: a=LDir(X9) and
A5: A=[PDir(Y),2];
  X is being_line by A3,AFF_1:36;
  then
A6: X9 '||' Y by A1,A2,A3,AFF_4:42,56;
  X9 is being_line by A3,AFF_1:36;
  hence thesis by A1,A4,A5,A6,Th29;
end;
