reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th9:
  for P1,P2 being IncProjStr st the IncProjStr of P1 = the
IncProjStr of P2 for A1 being Subset of the Points of P1 for A2 being Subset of
the Points of P2 st A1 = A2 for L1 being LINE of P1, L2 being LINE of P2 st L1
  = L2 holds A1 on L1 implies A2 on L2
proof
  let P1,P2 be IncProjStr;
  assume
A1: the IncProjStr of P1 = the IncProjStr of P2;
  let A1 be Subset of the Points of P1, A2 be Subset of the Points of P2;
  assume
A2: A1 = A2;
  let L1 be LINE of P1, L2 be LINE of P2;
  assume that
A3: L1 = L2 and
A4: A1 on L1;
  thus A2 on L2
  proof
    let A be POINT of P2;
    consider B being POINT of P1 such that
A5: A = B by A1;
    assume A in A2;
    then B on L1 by A2,A4,A5;
    then [A,L2] in the Inc of P2 by A1,A3,A5;
    hence thesis;
  end;
end;
