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;
