reserve X for set;
reserve CS for non empty CollStr;
reserve a,b,c for Point of CS;
reserve CLSP for CollSp;
reserve a,b,c,d,p,q,r for Point of CLSP;
reserve i,j,k for Element of NAT;
reserve CLSP for proper CollSp;
reserve a,b,c,p,q,r for Point of CLSP;
reserve P,Q for LINE of CLSP;

theorem Th17:
  P c= Q implies P = Q
proof
  assume
A1: P c= Q;
  Q c= P
  proof
    let r be object;
    consider p,q such that
    p<>q and
A2: P = Line(p,q) by Def7;
    assume
A3: r in Q;
    then reconsider r as Point of CLSP by Lm9;
    p in P & q in P by A2,Th10;
    then p,q,r are_collinear by A1,A3,Th16;
    hence thesis by A2;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
