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 Th17:
  {A,B,C} is linear implies {A,B,C,D} is planar
proof
  given L such that
A1: {A,B,C} on L;
  {A,B} \/ {C} on L by A1,ENUMSET1:3;
  then
A2: {A,B} on L by Th8;
  consider P such that
A3: {A,B,D} on P by Def12;
  {A,B} \/ {D} on P by A3,ENUMSET1:3;
  then
A4: {A,B} on P by Th11;
  assume
A5: not {A,B,C,D} is planar;
  then A <> B by Th16;
  then L on P by A2,A4,Def14;
  then
A6: {A,B,C} on P by A1,Th14;
  then
A7: C on P by Th4;
A8: D on P by A3,Th4;
  A on P & B on P by A6,Th4;
  then {A,B,C,D} on P by A7,A8,Th5;
  hence contradiction by A5;
end;
