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 Th8:
  F on L & A on L iff F \/ {A} on L
proof
  thus F on L & A on L implies F \/ {A} on L
  proof
    assume
A1: F on L & A on L;
    let C be POINT of S;
    assume C in F \/ {A};
    then C in F or C in {A} by XBOOLE_0:def 3;
    hence thesis by A1,TARSKI:def 1;
  end;
  assume
A2: F \/ {A} on L;
  hence F on L by Th6,XBOOLE_1:7;
  {A} c= F \/ {A} by XBOOLE_1:7;
  then {A,A} c= F \/ {A} by ENUMSET1:29;
  then {A,A} on L by A2;
  hence thesis by Th1;
end;
