reserve S for IncStruct;
reserve A,B,C,D for POINT of S;
reserve L for LINE of S;
reserve P for PLANE of S;
reserve F,G for Subset of the Points of S;
reserve a,b,c for Element of {0,1,2,3};
reserve S for IncSpace;
reserve A,B,C,D,E for POINT of S;
reserve K,L,L1,L2 for LINE of S;
reserve P,P1,P2,Q for PLANE of S;
reserve F for Subset of the Points of S;

theorem Th26:
  not A on L implies ex P st for Q holds A on Q & L on Q iff P = Q
proof
  assume
A1: not A on L;
  consider B,C such that
A2: B <> C and
A3: {B,C} on L by Def8;
  consider P such that
A4: {B,C,A} on P by Def12;
  take P;
  let Q;
  thus A on Q & L on Q implies P = Q
  proof
    assume that
A5: A on Q and
A6: L on Q;
    {B,C} on Q by A3,A6,Th14;
    then B on Q & C on Q by Th3;
    then
A7: {B,C,A} on Q by A5,Th4;
    not {B,C,A} is linear by A1,A2,A3,Th18;
    hence thesis by A4,A7,Def13;
  end;
A8: {B,C} \/ {A} on P by A4,ENUMSET1:3;
  thus thesis by A2,A3,A8,Def14,Th9;
end;
