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 Th5:
  {A,B,C,D} on P iff A on P & B on P & C on P & D on P
proof
  thus {A,B,C,D} on P implies A on P & B on P & C on P & D on P
  proof
A1: C in {A,B,C,D} & D in {A,B,C,D} by ENUMSET1:def 2;
A2: A in {A,B,C,D} & B in {A,B,C,D} by ENUMSET1:def 2;
    assume {A,B,C,D} on P;
    hence thesis by A2,A1;
  end;
  assume
A3: A on P & B on P & C on P & D on P;
  let E be POINT of S;
  assume E in {A,B,C,D};
  hence thesis by A3,ENUMSET1:def 2;
end;
