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 Th36:
  A // C iff ex a,b,c,d st a<>b & c <>d & a,b // c,d & A=Line(a,b)
  & C=Line(c,d)
proof
  thus A // C implies
    ex a,b,c,d st a<>b & c <>d & a,b // c,d & A=Line(a,b) & C=Line(c,d)
  proof
    assume A // C;
    then consider a,b such that
A1: A=Line(a,b) and
A2: a<>b and
A3: a,b // C;
    ex c,d st c <>d & C=Line(c,d) & a,b // c,d by A3;
    hence thesis by A1,A2;
  end;
    given a,b,c,d such that
A4: a<>b and
A5: c <>d and
A6: a,b // c,d and
A7: A=Line(a,b) and
A8: C=Line(c,d);
    a,b // C by A5,A6,A8;
    hence thesis by A4,A7;
end;
