theorem
  not {A,B,C} is linear implies ex P st for Q holds {A,B,C} on Q iff P = Q
proof
  assume
A1: not {A,B,C} is linear;
  consider P such that
A2: {A,B,C} on P by Def12;
  take P;
  thus thesis by A1,A2,Def13;
end;
