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 Th37:
  for A, C be being_line Subset of AS st
    a in A & b in A & c in C & d in C & a<>b & c<>d holds
      (A // C iff a,b // c,d)
proof
  let A, C be being_line Subset of AS;
  assume that
A1: a in A and
A2: b in A and
A3: c in C and
A4: d in C and
A5: a<>b and
A6: c <>d;
  thus A // C implies a,b // c,d
  proof
    assume A // C;
    then consider p,q,r,s such that
A7: p<>q and
A8: r<>s and
A9: p,q // r,s and
A10: A=Line(p,q) and
A11: C=Line(r,s) by Th36;
    p,q // a,b by A1,A2,A7,A10,Th21; then
A12: a,b // r,s by A7,A9,Th4;
    r,s // c,d by A3,A4,A8,A11,Th21;
    hence thesis by A8,A12,Th4;
  end;
A13: C=Line(c,d) by A3,A4,A6,Lm6;
    assume
A14: a,b // c,d;
    A=Line(a,b) by A1,A2,A5,Lm6;
    hence thesis by A5,A6,A14,A13,Th36;
end;
