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 Th42:
  a,b // A & A // C implies a,b // C
proof
  assume that
A1: a,b // A and
A2: A // C;
  consider p,q,c,d such that
A3: p<>q and
A4: c <>d and
A5: p,q // c,d and
A6: A=Line(p,q) and
A7: C=Line(c,d) by A2,Th36;
A8: q in A by A6,Th14;
A9: A is being_line by A2;
  p in A by A6,Th14;
  then a,b // p,q by A1,A3,A8,A9,Th26;
  then a,b // c,d by A3,A5,Th4;
  hence thesis by A4,A7;
end;
