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
  A <> B & A <> C & {A,B,C} is linear implies Line(A,B) = Line(A,C)
proof
  assume
A1: A <> B;
  then
A2: {A,B} on Line(A,B) by Def19;
  then
A3: A on Line(A,B) by Th1;
  assume
A4: A <> C;
  assume {A,B,C} is linear;
  then C on Line(A,B) by A1,A2,Th18;
  then {A,C} on Line(A,B) by A3,Th1;
  hence thesis by A4,Def19;
end;
