reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;
reserve A,C,D,K for Subset of AS;

theorem
  a in A & b in A & A // C implies a,b // C
proof
  assume that
A1: a in A and
A2: b in A and
A3: A // C;
A4: C is being_line by A3,Th35;
  now
    consider p,q such that
A5: p in C and
A6: q in C and
A7: p<>q by A4,Th18;
A8: C=Line(p,q) by A4,A5,A6,A7,Lm6;
    a,b // p,q by A1,A2,A3,A5,A6,Th38;
    hence thesis by A7,A8;
  end;
  hence thesis;
end;
