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;

theorem Th4:
  {A,B,C} on P iff A on P & B on P & C on P
proof
  thus {A,B,C} on P implies A on P & B on P & C on P
  proof
A1: C in {A,B,C} by ENUMSET1:def 1;
A2: A in {A,B,C} & B in {A,B,C} by ENUMSET1:def 1;
    assume {A,B,C} on P;
    hence thesis by A2,A1;
  end;
  assume
A3: A on P & B on P & C on P;
  let D be POINT of S;
  assume D in {A,B,C};
  hence thesis by A3,ENUMSET1:def 1;
end;
