theorem
  a,b // A & a,b // C & a<>b implies A // C
proof
  assume that
A1: a,b // A and
A2: a,b // C and
A3: a<>b;
A4: C is being_line by A2;
  then consider p,q such that
A5: p<>q and
A6: p in C and
A7: q in C and
A8: a,b // p,q by A2,Th29;
A9: A is being_line by A1;
  then consider c,d such that
A10: c <>d and
A11: c in A and
A12: d in A and
A13: a,b // c,d by A1,Th29;
  c,d // p,q by A3,A13,A8,Th4;
  hence thesis by A9,A4,A10,A11,A12,A5,A6,A7,Th37;
end;
