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);
reserve A,K,M,N,P,Q for LINE of IncProjSp_of(AS);

theorem Th38:
  for a,b being Element of the Points of ProjHorizon(AS), A,K
  being Element of the Lines of ProjHorizon(AS) st a on A & a on K & b on A & b
  on K holds a=b or A=K
proof
  let a,b be Element of the Points of ProjHorizon(AS), A,K be Element of the
  Lines of ProjHorizon(AS) such that
A1: a on A and
A2: a on K and
A3: b on A and
A4: b on K;
  consider Y9 such that
A5: b=LDir(Y9) and
A6: Y9 is being_line by Th14;
  consider X9 such that
A7: K=PDir(X9) and
A8: X9 is being_plane by Th15;
A9: Y9 '||' X9 by A4,A5,A6,A7,A8,Th36;
  consider Y such that
A10: a=LDir(Y) and
A11: Y is being_line by Th14;
  assume a<>b;
  then
A12: not Y // Y9 by A10,A11,A5,A6,Th11;
  consider X such that
A13: A=PDir(X) and
A14: X is being_plane by Th15;
A15: Y9 '||' X by A3,A5,A6,A13,A14,Th36;
A16: Y '||' X9 by A2,A10,A11,A7,A8,Th36;
  Y '||' X by A1,A10,A11,A13,A14,Th36;
  then X '||' X9 by A11,A6,A14,A8,A12,A16,A15,A9,Th5;
  hence thesis by A13,A14,A7,A8,Th13;
end;
