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 Th50:
  A on P implies ex L,L1,L2 st L1 <> L2 & L1 on P & L2 on P & not
  L on P & A on L & A on L1 & A on L2
proof
  consider B,C such that
A1: {B,C} on P and
A2: not {A,B,C} is linear by Th46;
  consider D such that
A3: not {A,B,C,D} is planar by A2,Th45;
  assume A on P;
  then
A4: {A} \/ {B,C} on P by A1,Th9;
  then
A5: {A,B,C} on P by ENUMSET1:2;
  take L3 = Line(A,D),L1 = Line(A,B),L2 = Line(A,C);
A6: A <> B by A2,Th15;
  then
A7: {A,B} on L1 by Def19;
A8: not {A,C,B} is linear by A2,ENUMSET1:57;
  then
A9: A <> C by Th15;
  then
A10: {A,C} on L2 by Def19;
  then B on L2 implies {A,C} \/ {B} on L2 by Th8;
  then B on L2 implies {A,C,B} on L2 by ENUMSET1:3;
  hence L1 <> L2 by A8,A7,Th1;
  {A,B} \/ {C} on P by A5,ENUMSET1:3;
  then {A,B} on P by Th11;
  hence L1 on P by A6,A7,Def14;
  {A,C,B} on P by A4,ENUMSET1:2;
  then {A,C} \/ {B} on P by ENUMSET1:3;
  then {A,C} on P by Th9;
  hence L2 on P by A9,A10,Def14;
  not {A,D,B,C} is planar by A3,ENUMSET1:63;
  then A <> D by Th16;
  then
A11: {A,D} on L3 by Def19;
  then L3 on P implies {A,D} on P by Th14;
  then L3 on P implies D on P by Th3;
  then L3 on P implies {A,B,C} \/ {D} on P by A5,Th9;
  then L3 on P implies {A,B,C,D} on P by ENUMSET1:6;
  hence not L3 on P by A3;
  thus thesis by A10,A7,A11,Th1;
end;
